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Môn: Đại số (MAT1093) 33 tài liệu

Trường: Trường Đại học Công nghệ, Đại học Quốc gia Hà Nội 591 tài liệu

Tác giả:

Profile

Mai Nguyệt

1 năm trước
Danh sách Quiz
/24
2 5x − y = 10
x + y + z = 1
x
1
− x
2
= 2
x
2
= 3
−x + y − z = 0
2y + z = 3
10z = 0
x − y = 4
2y + z = 6
3z = 6
x
1
+ x
2
+ x x
3
+
4
= 10
x
2
+ 3x x
3
+ 4
4
= 15
x + 2 = 7y
2x + y = 8
x + 2y = 0
x + y = 6
3x − 2y = 8
x + y + z = 6
2x − y + z = 3
3x − z = 0
x + y + z = 2
−x + 3y + 2z = 8
4x + y = 4
3 2x − y + 4 = 1z
x + y − 2z = 3
2 3x − y + 6 = 8z
5 3x − y + 2 = 3z
2x + 4y − z = 7
x − 11y + 4z = 3
x + y + z + w = 6
2x + 3 = 0y − w
−3x + 4 = 4y + z + 2w
x + 2y − z + w = 0
4x + 3 + 17 = 0y z
5x + 4 + 22 = 0y z
4x + 2 + 19 = 0y z
2
x
+
3
y
= 0
3
x
−
4
y
= −
25
6
2
x
+
1
y
−
3
z
= 4
4
x
+
2
z
= 10
−
2
x
+
3
y
−
13
z
= −8
k
4x + ky = 6
kx + y = −3
x + 2 = 6y + kz
3x + 6 = 4y + 8z
x + y + kz = 3
x + ky + z = 2
kx + y + z = 1
−1 3
0
−2
1 2 3
0 1 1
0 0 0
1 2 3
0 1 6
0 1 9
1 2 3 4 5
0 4 3 2 1
0 0 1 1 1
0 0 0 9 9
1 2 3 4 5
1 0 0 0 0
0 1 0 0 0
0 0 1 0 1
0 0 0 9 9
1 0 0
0 1 2
1 −1 0 3
0 1 −2 1
0 0 1 −1
1 2 1 0
0 0 1 1−
0 0 0 0
2 1 −1 3
1 −1 1 0
0 1 2 1
2 1 1 0
1 −2 1 2−
1 0 1 0
1 2 0 1 4
0 1 2− − −1 −3
0 0 1 2 1
0 0 0 −1 −4
− −3x + 5y = 22
3x + 4y = 4
4x − 8y = 32
x − 3z = −2
3 2x + y − z = 5
2x + 2y + z = 4
x + y − 5z = 3
x − 2z = 1
2x − y − z = 0
2x − y + 3 = 24z
2y − z = 14
7 5x − y = 6
x + 2y + z = 8
−3x − 6y − 3z = −21
2x + y − z + 2 6w = −
3x + 4y + w = 1
x + 5y + 2z + 6w = −3
5x + 2y − z − w = 3
A
:=
1 k 2
−3 4 1
−3 6 −6
.
A k
A k
B
:=
2 −1 3
−4 2 k
4 −2 6
.
B k
B k
1 0 1 1
0 1 2 1
0 0 0 1
−1 2 1
0 1 0
0 0 1
−1 1 2 1
2 3 1 2−
5 4 2 4
2 3 1 10
2 −3 −3 22
4 −2 3 2−
2x + 3 = 3z
4x − 3y + 7 = 5z
8x − 9y + 15 = 10z
2x + 3y + 3 = 3z
6x + 6 + 12 = 13y z
12x + 9y − z = 2
2x + 6 9z = −
3x − 2y + 11z = −16
3x − y + 7z = −11
2x + 5y − 19z = 34
3x + 8 = 54y − 31z
4x + 12y − 7z − 20w = 22
3x + 9y − 5z − 28w = 30
x + 2y + 6z = 1
2x + 5 + 15y z = 4
3x + y + 3 6z = −
2x + y + 2 = 4z
2x + 2 = 5y
2x − y + 6 = 2z
2x + y + z + 2 1w = −
5x − 2 3y + z − w = 0
−x + 3y + 2z + 2w = 1
3 5x + 2y + 3z − w = 12
k
x − y + 2z = 0
−x + y − z = 0
x + ky + z = 0
a, b, c
x + 2y = 3
ax + by = −9
x + y = 2
y + z = 2
x + z = 2
ax + by + cz = 0
x + y = 0
y + z = 0
x + z = 0
ax + by + cz = 0
2x − y + z = a
x + y + 2z = b
3y + 3z = c
A
+ B, A − B, 2A, B +
1
2
A
A, B
A
=
6 −1
2 4
−3 5
, B =
1 4
−1 5
1 10
A
=
2 1 1
− −
1 1 4
, B =
2 −3 4
−
3 1 −2
A
=
3 2 1−
2 4 5
0 1 2
, B =
0 2 1
5 4 2
2 1 0
A
=
2 3 4
0 1 1−
2 0 1
, B =
0 6 2
4 1 0
−1 2 4
AB
t
, BA , A B
t t
A, B
X
A
=
−4 0
1 −5
−3 2
, B =
1 2
−2 1
4 4
AC = BC A = B
A
=
0 1
0 1
, B =
1 0
1 0
, C =
2 3
2 3
A
=
1 2 3
0 5 4
3 −2 1
, B =
4 −6 3
5 4 4
−1 0 1
, C =
0 0 0
0 0 0
4 −2 3
4
x y
z
−1
= 2
y z
−
x 1
+ 2
4 x
5
−x
w x
y x
=
−4 3
2
−1
+ 2
y w
z x
1 2
3 5
A =
1 0
0 1
2 −1
3
−2
A =
1 0
0 1
1 2
3 4
A =
6 3
19 2
A
2 1
3 1
=
3 17
4
−1
AB = BA
A
=
x y
z w
, B =
1 1
−
1 1
b A A, b
A
=
1 −1 2
3
−3 1
, b =
−1
7
A
=
1 2 4
−1 0 2
0 1 3
, b =
1
3
2
A
=
1 1 5−
1 0 1−
2 1− −1
, b =
3
1
0
A
=
−3 5
3 4
4 −8
, b =
−22
4
32
X
=
1
0
1
, Y =
1
1
0
, Z =
2
−1
3
, W =
1
1
1
, O =
0
0
0
a b Z = aX + bY
a b W = aX + bY
aX + bY + cW O= a = b = c = 0
a, b c aX + bY + cZ O=
A
5
A
10
A
=
1 0 0
0 −1 0
0 0 2
.
B A A, B
A
=
1 2
3 4
, B =
−2 1
3
2
−
1
2
A
=
−2 2 3
1 −1 0
0 1 4
, B =
1
3
− −4 5 3
− −4 8 3
1 2 0
A
=
1 −1
2 3
, B =
1
5
3 1
−
2 1
A
=
2 −17 11
− −1 11 7
0 3 2−
, B =
1 1 2
2 4 3−
3 6 5−
1 2
3 7
sin θ cos θ
−
cos θ sin θ
1 1 1
3 5 4
3 6 5
1 2 2
3 7 9
− −1 4 −7
1 2 1−
3 7 −10
7 16 −21
10 5 7−
−5 1 4
3 2 2−
1 1 2
3 1 0
−2 0 3
3 2 5
2 2 4
−4 4 0
0.1 0 2 0 3. .
−0.3 0 2 0 2. .
0.5 0 5 0 5. .
( )
AB
−1
, ( ) )A
t −1
, A
−2
, (2B
−1
A
−1
=
2 5
−
7 6
, B
−1
=
7 −3
2 0
A
−1
=
1
−
1
2
3
4
3
2
1
2
−2
1
4
1
1
2
, B
−1
=
2 4
5
2
−
3
4
2
1
4
1
4
1
2
2
A
−1
=
−
2
7
1
7
3
7
2
7
, B
−1
=
5
11
2
11
3
11
−
1
11
A
−1
=
1 −4 2
0 1 3
4 2 1
, B
−1
=
6 5 3−
−2 4 −1
1 3 4
x
A
= A
−1
A
=
3 x
− −
2 3
A
=
2 x
− −
1 2
x A A
A
(2
A)
−1
=
1 2
3 4
(4
A)
−1
=
2 4
−
3 2
Ax b=
A
−1
5x + 4y = 2
− −x + y = 22
−x + y + 2z = 1
2x + 3 2y + z = −
5x + 4 + 2y z = 4
3x + 2y = 1
x + 4y = −3
x + y + 2z = 0
x − y + z = −1
2x + y + z = 2
1 2
3 4
λ − 3 2
4
λ − 1
1 2 3−
6 5 4
1 −3 2
2 4 3−
1 3 6
− −8 7 4
1 0 2
−1 1 4
2 0 3
1 4 2−
3 2 0
−1 4 3
2 −1 3
1 4 4
1 0 2
x y 1
2 3 1
0 −1 1
x y 1
−2 −2 1
1 5 1
2 6 6 2
2 7 3 6
1 5 0 1
3 7 0 7
1 4 3 2
−5 6 2 1
0 0 0 0
3 −2 1 5
5 3 0 6
4 6 4 12
0 2 −3 4
0 1 −2 2
3 0 7 0
2 6 11 12
4 1 −1 2
1 5 2 10
5 2 0 0 2−
0 1 4 3 2
0 0 2 6 3
0 0 3 4 1
0 0 0 0 2
x
x + 3 1
2
x + 2
= 0
x − 2 −1
−
3 x
= 0
x 2 0
0 x + 1 2
0 1 x
= 0
x 0 1
0 x 3
2 2 2x −
= 0
1 1 1
a b c
a
2
b
2
c
2
= (a − b)( )( )b − c c − a
a 1 1 1
1 a 1 1
1 1 1a
1 1 1 a
= (
a + 3)(a − 1)
3
a + b a a
a a + b a
a a a b+
=
b
2
(3 )a + b
1 + a 1 1
1 1 + 1b
1 1 1 + c
=
abc
1 +
1
a
+
1
b
+
1
c
(a, b, c 6= 0)
1 7 3−
1 3 1
4 8 1
1 1 1
2 −1 −2
1 −2 −1
3 1− −3
−1 − −4 2
3 1− −1
4 5 2−
3 4 3
−2 1 4
4 −7 9 1
6 2 7 0
3 6 −3 3
0 7 4 −1
9 −4 2 5
2 7 6 −5
4 1 −2 0
7 3 4 10
0 −3 8 2
8 1 −1 6
−4 6 0 9
−7 0 0 14
| | |A , |B , AB, |AB| | |A||B| = AB|
A
=
−1 2 1
1 0 1
0 1 0
, B =
−1 1 0
0 2 0
0 0 3
A
=
2 0 1
1 −1 2
3 1 0
, B =
2 −1 4
0 1 3
3 −2 1
A
=
2 0 1 1
1 −1 0 1
2 3 1 0
1 2 3 0
, B
=
1 0 1 1−
2 1 0 2
1 1 1 0−
3 2 1 0
A
=
3 2 4 0
1 −1 2 1
0 0 3 1
−1 1 1 0
, B
=
4 2 −1 0
1 1 2 1−
0 0 2 1
−1 0 0 0
| | |
A
t
| | |, A
2
, |AA
t
, 2A|, |A
−1
|
A
=
2 0 5
4 −1 6
3 2 1
A
=
1 5 4
0 −6 2
0 0 3−
1 2
3 4
−1 0
0 4
1 0 0
0 2 6
0 −4 −12
1 2 3
0 1 1−
2 2 2
− − −3 5 7
2 4 3
0 1 1−
0 1 1
1 2 3
− − −1 1 2
x + 2 = 5y
−x + y = 1
4x − y − z = 1
2x + 2 + 3y z = 10
5 2x − 2y − z = −1
4x − 2y + 3 2z = −
2x + 2 + 5y z = 16
8 2x − 5y − z = 4
3x + 3 = 1y + 5z
3x + 5 = 2y + 9z
5x + 9 + 17 = 4y z
2x + 3y + 5 = 4z
3x + 5y + 9 = 7z
5x + 9 + 17 = 13y z
2x + y + 2 = 6z
−x + 2y − 3z = 0
3x + 2 = 6y − z
k
k
kx + (1 − k)y = 1
(1 − k)x + ky = 3
(0 0) (2 0) (0 3);, , , , , (1 1) (2 4) (4 2), , , , , .
(1 2) (3 4) (5 6);, , , , , (−1, 0) (1 1) (3 3), , , , .
(−4, 7) (2 4);, , (−2 2, 3), (− , −4).
(1 1) (0 0) (2 1) 2);, 1, , , 0, , , 1, − , (−1, 1, (3 1) (4 4) (1 1) (0 1), −1, , , −4, , , 1, , , 0, .
(−4, 1 1 3 0, 0) (0, , , 2) (4, , , −1) (0, , , 1);
(1 3) 1) (0 5) (2 11);, 2, , (−1, 0, , , −2, − , , 6,
(0 1) (0 0) (1 0) (2 2);, 0, − , , −1, , , 1, , , 1,
(1 7) 6) (4 2) (3 4), 2, , (−3 6, , , , 4, , , 3, .
(1 1) 7) (2 3);, −2, , (−1 1, − , , , −1,
(0 0) (1 0) (2 2);, −1, , , 1, , , 1,
(0 0) (1 0) (0 1);, 0, , , −1, , , 1, −
(1 7) (4 2) (3 4), 2, , , 4, , , 3, .
= 0 = 1 = 2A B C = 3 = 4 = 5 = 6 = 7 = 8D E F G H
I = 9 J = 10 K = 11 L = 12 M = 13 N = 14 O = 15 P = 16 Q = 17
R = 18 S = 19 T = 20 U = 21 V = 22 W = 23 X = 24 Y = 25 Z = 26.
A
=
1 1 1
0 1 1
1 1 2
.
D
T H U B A
20 8 21 0 2 1
,
D =
20 8 21
0 2 1
.
DA
=
20 8 21
0 2 1
1 1 1
0 1 1
1 1 2
=
41 49 70
1 3 4
A
B
=
27 42 56
29 30 55
A
A
=
1 1 1
1 2 2
1 1 2
.
B
=
33 51 60
30 31 56
A
u = (1, 2, ,3), v = (2, 2 −1), w = (4, 0, −4).
v v v
−u, u−v+2w, 2u+4 −w, 5u−3 −
1
2
w.
z 2z − 3u = w.
z 2u + v − w + 3z = 0.
R
4
M
2,3
M
1,4
M
2,2
P
3
P
3
3.
1 f(x) = ax b+
1 f(x) = ax
a 6= 0
{ }
(x, y) ∈ R
2
: x > 0
{
(x, y) ∈ R
2
: x > 0 0, y > }
{
(x, y) ∈ R
2
: x = 2y}
a b
c
0
;
a b
c
1
;
2 × 2 0
2 × 2 0
R
2
( (x
1
, y
1
) ⊎ x
2
, y
2
) = ( )x
1
+ x
2
, y
1
+ y
2
c ∗ (x, y) = (cx, y)
( (x
1
, y
1
) ⊎ x
2
, y
2
) = ( )x
1
+ x
2
, y
1
+ y
2
c
∗ (x, y) = (
√
cx,
√
cy)
( (x
1
, y
1
) ⊎ x
2
, y
2
) = (x
1
, 0) c ∗ (x, y) = (cx, cy)
( (x
1
, y
1
) ⊎ x
2
, y
2
) = ( )x x
1 2
, y
1
y
2
c ∗ (x, y) = (cx, cy)
R
3
( (x
1
, y , z
1 1
) ⊎ x
2
, y , z
2 2
) = (0, 0, 0) c ∗ (x, y, z cx, cy, cz) = ( )
( (x
1
, y , z
1 1
) ⊎ x
2
, y , z
2 2
) = (x
1
+ x
2
+ 1 + 1 + 1), y
1
+ y
2
, z
1
+ z
2
c ∗ (x, y, z cx, cy, cz) = ( )
W V
W
= {(x, y, z, w) ∈ V = R
4
: w = 0}
W
= {(x, y, z) ∈ V = R
3
: z = 2x + 3y}
W
=

0 b
a
0
∈ V = M
2,2
: a, b ∈ R
W
=
a b
a + b 0
0 c
∈ ∈V = M
3,2
: a, b, c R
W
= {(x, y, z) ∈ V = R
3
: z = 1}
W
= {(x, y, z x, y, z) ∈ V = R
3
: ∈ Z}
W
= {(x, y, z) ∈ V = R
3
: x, y, z ∈ Q}
W
= {(x, y) ∈ V = R
2
: y = x
2
}
W
= {(x, y, z) ∈ V = R
3
: z = 1/x, x 6 }= 0
W
= {(x, y, z) ∈ V = R
3
: x
2
+ y
2
= z
2
}
W
= {x ∈ V = R
n
: Ax = 0}
A
m × n
W
= {x ∈ V = R
n
: Ax = b 6 }= 0
A
m × n
u, v S
S = {(2, ,−1, , , , , , ,3) (5, 0 4)}, u = (1, 1 −1), v = (−1 −2 2), w = (1 −8 12)
S = {(1, 2, , , , , , , ,−2) (2, −1 1)}, u = (1 −5 −5), v = (−4, −3 3), w = (−2 −6 6)
u = (1, 2), v .= (1, −1)
u v
(2 1), (3 0), (0 3), (1 1), −
z u, v w
z = (10, 1, 4) u = (2, , ,3, 5), v = (1 2 4), w = (−2, 2, 3)
z = (−1, 7, 2) u = (1, , , , ,3 5), v = (2 −1 3), w = (−3 2, −4)
z = (0, ,5 3, 0) u = (1, ,1, , ,2, 2), v = (2 3 5 6), w = (−3, ,1 −4, 2)
z = (2, ,5, −4 0) u = (1, , , , , , ,3 2, 1), v = (2 −2 −5 4), w = (2 −1 3, 6)
{ }(0 0) (1 2), , ,
{ }(1 0) (1 1) (2, , , , , −1)
{ }(1, −4, 1) (6 2), , 3,
{(6 1), 2, , (−1 3, , 2)}
{(1 1) (2 2) (1 3), 1, , , 2, , , 2, }
{(−4, −3, 4) (1, , −2, 3) (6 0), , 0, }
{(1 0) (0 0) (0, 0, , , 4, , , 0, −6) (1, , 5, −3)}
{(0 1) (0 1) (0 1) (1 1), 0 0, , , , 0, 1, , , 1, 1, , , 1, 1, }
0
{ }(3 4), , (−1, 1) (2 0), ,
{ }(2 4), , (−1, −2) (0 6), ,
{(1 1) (1 0) (0 1) (0 1), 1, , , 1, , , 1, , , 0, }
{(1 4) (1 2) (1 6), 2, 3, , , 0 1, , , , 4, 5, }
t
{ }(t, 1, 1), (1, t, 1), (1, 1, t))
{ }(t, 1, 1), (1, 0, 1), (1, 1, 3t))
{(t, 0 0, 0) (0 0) (0, , 1, , , , 1))}
{( ) (t, t, t , (t, 1, 0), t, 0, 1))}
A
=
2 −3
4 1
, B =
0 5
1
−2
.
A B
6 2
9 11
0 0
0 0
6 −19
10 7
−2 28
1
−11
1 −1
4 5
,
4 3
−
2 3
,
1 −8
22 23
P
2
{ }
2 − x, 2x − x
2
, 6 − 5x + x
2
{ }
x
2
− 1 2, x + 5
{
x x
2
+ 3x + 1, 2
2
+ x − 1, 4x}
{
x
2
, x
2
+ 1}
R
2
{(2 1), , (−1, 2)}
{(1 1), }
{ }(1, −2), (−3, 9)
{(1 3), , (−2, −6) (3 9), , }
{ }(−1, 4) (4, , −1) (1 1), ,
{ }(1, −2), (−2, 1) (1 1), ,
R
3
{(4 3), 7, , (−1, 2, 6) (2, , −3, 5)}
{(6 6) (3, 7, , , 2, −4) (1, , −3, 2)}
{(−2, 5 6, 0) (4, , , 3)}
{ }(1 0) (1 1) (0 1), 1, , , 0, , , 1,
{ }(1, −2, 0) (0 1), , 0, , (−1 2, , 0)
{ }(1 3) (2, 0, , , 0, −1) (4 5) (2 6), , 0, , , 0,
{
1, x , x
2 2
+ 2}
P
2
{
x
2
− − −2x, x
3
+ 8, x
3
x
2
, x
2
4}
P
3
R
4
M
2,3
M
4,1
M
2,2
R
2
{(1 2) (1 0) (0 1), , , , , }
{(1 1), }
{(−4, 5) (0 0), , }
{(1, − −2), ( 1, 2) (2 4), , }
{ }(1 2) (3 4), , ,
R
3
{(1 0) (4 2), 3, , , 1, , (− −2, 5, 2)} {(1 1) (1 3), 1, , , 2, } {(1 3) (0 2) (0 6), 5, , , 1, , , 0, }
P
2
{
1, 2x, x
2
− 4 5, x}
{
1 − −x, 1 x
2
, 3x
2
− 2x − 1}
{ }
6x − 3, 3x
2
, 1 − 2x − x
2
{ }
x − 1, x
2
− 1, 1 − 2x − x
2
R
3
u = (8, 3, 8)
{(4 2) (0 2) (0 2), 3, , , 3, , , 0, } {(1 1) (1 0) (1 0), 1, , , 1, , , 0, } {(1 7) (3 1) (2 2), 4, , , 0, , , 1, }
M
2,2

2 0
0 3
,
1 4
0 1
,
0 1
3 2
,
0 1
2 0


1 2
−
5 4
,
2 −7
6 2
,
4 −9
10 12
,
3 −7
1 2

2
3
R
2
{(1 0) (0 1) (1 2), , , , , }
R
3
{(1, 3, − −2), ( 4 1, , 1), (−2, 7, −3) (2 1), , 1, }
R
2
{(1 1), }
R
3
{(1 2) (0 1), 0, , , 1, }
R
2
W = {(2t, t) : t ∈ R} W = {(0, t) : t ∈ R}
R
3
W = {(2t, t, −t) : t ∈ R} W = {(2s − t, s, t) : t, s ∈ R}
R
4
W = {(2s − t, s, t, s t, s) : ∈ R}
W = {(5t, −3t, t, t) : t ∈ R}
W = {(t, 2s − 3t, w, t) : t, s, w ∈ R}
1 1
2 −3 1
5 10 6
8 −7 5
1 2 3
1
−3 2
1 2 3
− −2 4 4 5
3 6 4−6 −
− −2 4 4 9
2 4 6−3 −
7 14 3−6 −
− −2 4 1 −2
2 4 2−2 −
R
3
S S
{(1 4), 2, , (−1 3 3, , 4) (2, , , 1)}
{(4, 2, −1) (1, , 2, −8) (0 2), , 1, }
{ }(1 2) (4 8) (1 1), 1, , , 4, , , 1,
{ }(1 2), 2, , (−1 0 1, , 0) (1, , , 1)
R
4
S S
{(2, 9, − − −2, 53), (− −3 3, 2, , 2) (8, , 3, −8, 17), (0, 3 0, , 15)}
{(2, 5, − −3, −2), ( 2, −3 2, , − −5) (1, , 3, −2, 2), ( 1, −5, 3, 5)}
Ax = 0 A
1 4 2
2 −1
1 3
1 2 3
1 0 0
1 2 3−
2 −1 4
4 3 2−
1 3 −2 4
0 1 −1 2
− −2 6 4 −8
−x + y + z = 0
−3x − y = 0
2 5x − 4y − z = 0
4x − y + 2 = 0z
2x + 3 = 0y − z
3x + y + z = 0
x − 2y + 3 = 0z
−3x + 6y − 9z = 0
x + 2 = 0y − 4z
−3x − 6y + 12 = 0z
3x + 3 + 15 + 11 = 0y z t
x − 3y + z + t = 0
2x + 3 + 11 = 0y z + 8t
Ax b=
x = x x
h
+
p
x
h
Ax = 0 x
p
Ax b=
x + 3y + 10z = 18
−2x + 7 + 32y z = 29
−x + 3y + 14z = 12
x + y + 2z = 8
3x − 6y + z = 12
− −7x + 14 + 4y z = 28
2x − 4y + 5z = 8
b A b
A
A
=
−1 2
4 0
, b =
3
4
A
=
−1 2
2
−4
, b =
2
4
A
=
1 3 0
−1 1 0
0 1 1
, b =
1
2
−3
x x B
B
= {(2, ,−1) (0, 1)}, [x]
B
= [4, 1]
t
B
= {(1, 0, , , , , , ,1) (1 1, 0) (0 1 1)}, [x]
B
= [2 3, 1]
t
B
= {(0, ,0, , , , , , , , ,0 1), ,(0 0 1 1) (0, 1 1 1) (1 1 1, , ,1)}, [x]
B
= [1 −2 3, −1]
t
x B
B = {(−6, , ,7), (4 −3)}, x = (−26 32)
B = {(8, , ,11 0), , , , ,(7 0 10) (1 4 6)}, x = (3, 19, 2)
B = {(9, , , , , , ,−3, , ,15 4) (3 0, 0 1), ,(0 −5 6 8) (3, , , , ,−4 2 −3)}, x = (0 −20 7 15)
B B
′
B = {(1, ,0), (0 1)}, B
′
= {(2, 4), (1, 3)}
B = {(1, ,1), (1 0)}, B
′
= {(1, 0), (0, 1)}
B = {(1, ,0, , ,0), ,(0 1 0), , , ,(0 0 1)}, B
′
= {(1 0 0), , ,(0 2 8) (6, 0, 12)}
B = {(1, 0, 0), , , , , ,(0 1 0) (0 0 1)}, B
′
= {(1, , , ,3, −1) (2, 7, ,−4) (2 9 −7)}
P B B
′
Q B
′
B
P
= Q
−1
[x]
B
B
= {(1, 3), , ,(−2 −2)}, B
′
= {(−12 0), ,(−4 4)}, [x]
B
′
= [−1, 3]
t
B
= {(2, ,−2) (6, , ,3)}, B
′
= {(1 1), (32 31)}, [x]
B
′
= [2, −1]
t
B
= {(1, ,0, ,2), , , , , , ,(0 1 3) (1 1 1)}, B
′
= {(2 1 1), , ,(1 0 0) (0, 2, 1)}, [x]
B
′
= [1, 2, −1]
t
B
= {(1, , ,1, , , , , , , , ,1) (1 −1 1) (0, 0 1)}, B
′
= {(2 2 0) (0, 1 1), ,(1 0 1)}, [x]
B
′
= [2, 3, 1]
t
P
2
:
p
= x
2
+ 11x + 4
p
= 3x
2
+ 114x + 13
p
= −2x
2
+ 5x + 1
p
= 4x
2
− 3x − 2
X M
3,1
X
=
0
3
2
X
=
2
−1
4
X
=
1
2
−1
X
=
1
0
−4
R
n
v = (2, ,0, −4 5)
v = (1, ,−3, −5 6, 2)
v = (8, 8, 6) u
u v
u v
u = (1, 2, ,−4 3) v = (5, 1, 2, 3)
u = (1, , ,−3 −5 4, 2) u = (2, , ,−1 −2, 3 1)
u = (0, 1, ,2 3) v = (1, 0, ,4 −1)
k
uk
2
= 4, k kv
2
= 10
hu, vi = −5 hu + v, 2u − vi
k
uk k
2
= 8, kv
2
= 6
hu, vi = 7 h3u − v, u − 3vi
u = (1, 1, −2) v = (1, ,−3 −2)
u = (1, 2, ,3 4) v = (4, 3, ,2 1)
u = (1, 1, 1) v = (0, 1, −2)
u = (−1, 1, ,−1 1) v = (1, ,2 3, 4)
u = (3, 1) v = (−2, 4)
u = (1, 0, ,1 0) v = (3, 3, ,3 3)
u
=
cos
π
6
, sin
π
6
v
=
cos
3π
4
, sin
3π
4
u
=
cos
π
3
, sin
π
3
v
=
cos
π
4
, sin
π
4
u = (2, 7)
u = (2, −1, 1)
u = (0, , ,0 −1 1)
u = (cos x, sin x, −1) v = (sin x, −cos x, 0)
u = (−sin x, cos x, 1) v = (sin x, −cos x, 0)
hu, vi, k ku , k kv , d(u, v)
u = (4, 3) v = (0, 5) hu, vi = 3u
1
v
1
+ u
2
v
2
u = (1, 1, 1) v = (2, 5, 2) hu, vi = u
1
v
1
+ 2 + 3u
2
v
2
u
3
v
3
hA, Bi, k kAk, Bk, d(A, B) hA, Bi = 2a
11
b
11
+a
12
b
12
+a
21
b
21
+2a
22
b
22
A
=
−1 3
4
−2
, B =
0 −2
1 1
;
A
=
1 −1
2 4
, B =
0 1
−
2 0
.
hu, vi
R
2
hu, vi = u
1
v
1
hu, vi = u
1
v
1
− u
2
v
2
h
u, vi = u
2
1
v
2
1
+ u
2
2
v
2
2
hu, vi = u
1
u
2
+ v v
1 2
A
=
0 3
2 1
, B =
−3 1
4 3
hA, Bi = 2a
11
b
11
+ a
12
b
12
+ a
21
b
21
+ 2 ;a
22
b
22
A
=
0 1
2
−1
, B =
1 1
2
−2
hA, Bi = a
11
b
11
+ 2 + 2a
12
b
12
+ a
21
b
21
a .
22
b
22
u = (−4, 3) v = (0, 5) hu, vi = 3u
1
v
1
+ u
2
v
2
u = (1, ,1 1) v = (2, −2, 2) hu, vi = u
1
v
1
+ 2u
2
v
2
+ u
3
v
3
proj
u
v proj
v
u
u = (1, 2) v = (2, 1) proj
u
v proj
v
u
R
2
u = (−1, 3) v = (4, 4) proj
u
v proj
v
u
R
2
u = (1, 3, −2) v = (0, ,−1 1)
u = (0, 1, ,3 −6) v = (−1, ,1 2, 2)
u = (5, ,1 4) v = (2, ,1 −1) u v hu, vi =
u
1
v
1
+ 2u
2
v
2
+ 3u
3
v
3
u v
w = (2, 7) hu, vi = u
1
v
1
+ 3u
2
v
2
w = (2, ,−1 1) hu, vi = 2u
1
v
1
+ 3u
2
v
2
+ u
3
v
3
w = (−1, ,1, −1 1) hu, vi = u
1
v
1
+ 3 + 3u
2
v
2
u
3
v
3
+ u
4
v
4
S = {(4, −1, , , , ,1) (−1 0 4), ,(−4 −17 −1)}
S
=
n
√
2
2
, 0,
√
2
2
,
−
√
6
6
,
√
6
3
,
√
6
6
,
√
3
3
,
√
3
3
, −
√
3
3
o
S
=
n
√
2
2
, 0 0, ,
√
2
2
,
0,
√
2
2
,
√
2
2
, 0
,
−
1
2
,
1
2
, −
1
2
,
1
2
o
x
S
=
n
√
5
5
,
2
√
5
5
,
−
2
√
5
5
,
√
5
5
o
x = (−3, 4)
S
=
n
3
5
,
4
5
, 0
,
−
4
5
,
3
5
, 0
, (0 1), 0,
o
x = (5, ,10 15)
S = {(1, −2, , , ,2), (2 2, ,1) (2 −1 −2)}
S = {(4, −3, ,0), (1, , ,2 0) (0 0, 4)}
S = {(1, , ,0, ,0), (1 1 1) (1, 1, −1)}
S = {(0, , ,1, ,2), (2 0 0) (1, 1, 1)}
S = {(0, , ,1, ,1), (1 1 0) (1, 0, 1)}
x x x x
1
+
2
− 3
3
− 2
4
= 0
2 2x x
1
−
2
− x
4
= 0
3 5 4x
1
+ x
2
− x
3
− x
4
= 0;
x x x x
1
+
2
− 3
3
+ 2
4
= 0
x x x x
1
+ 2
2
− 3
3
+ 4
4
= 0
2 6x
1
+ x
2
− x
3
+ 2 = 0;x
4
x
1
+ x x x
2
−
3
−
4
= 0
2 2x
1
+ x
2
− x
3
− 2x
4
= 0;
x
1
− x x
2
+ x
3
+
4
= 0
x x x x
1
− 2
2
+
3
+
4
= 0;
x x x
1
− 2
2
+
3
= 0.
V
1
= span{(2, 1, , ,−1) (0 1, 1)} V
2
= span{(−1, 2, 0)}
V
1
= span{(0, ,0, , , , ,2 1) (0 0 1 −2)} V
2
= span{(3, , , , ,2 0, ,0) (0 1 −2 0)}
V = span{(1, , , , ,2 3) (1 1 1)}
V = span{(1, , , ,2 0, 0), (0 1, 0 1)}
v V
V = span{(0, , , ,0 −1, 1), (0 1, 1 1)} v = (1, ,0 1, 1)
V = span{(1, , ,0 1), ,(0 1 1)} v = (2, 3, 4)
T
: R R
2
→
2
, T (x, y) = (x, 1)
T
: R R
3
→
3
, T (x, y, z) = (x + y, x y, z− )
T
: R R
3
→
3
, T (x, y, z) = (x + 1, y , z+ 1 + 1)
T : M
2,2
→ R, T (A) = |A| = det A;
T
: M
3,3
→ M
3,3
, T (A) =
0 0 1
0 1 0
1 0 0
A;
T
: M
2,2
→ M
2,2
, T (A A) =
T
T
: R
3
→ R
3
T (1, ,0 0) = (2, , , ,4 −1), T (0 1, ,0) = (1 3 −2)
T (0, ,0, 1) = (0 −2, 2)
T (0, 3, −1)
T (2, ,−1 0)
T
: R
3
→ R
3
T (1, , , ,1, , ,1) = (2 0 −1), T (0 −1, 2) = (−3 2 −1)
T (1, , , ,0 1) = (1 1 0)
T (2, 1, 0)
T (2, ,−1 1)
T
: R R
2
→
3
A
=
1 2
−2 4
−2 2
,
T (2, 4)
T
−1
(−1, ,2 2).
T
: R
3
→ R
3
, T (x, y, z) = (x, 0, z)
T
: R
3
→ R
3
, T (x, y, z z, y, x) = ( )
ker T imT T T(v) = Av
A
=
1 −1 2
0 1 2
;
A
=
1 2
− −1 2
1 1
;
A
=
4 1
0 0
2 −3
;
A
=
1 1 0 0
0 0 1 1
.
ker T T
T
T
: R R
2
→
2
, T (x, y) = (x + 2y, x − 2y)
T
: R R
3
→
3
, T (x, y, z) = (2x − 3y, x y, z− )
T
: R R
3
→
3
, T (x, y, z) = (0, 0, 0)
T
: R R
4
→
2
, T (x
1
, x , x , x , x
2 3 4
) = (x x
1
+
2 3
+ x
4
)
T = T
2
◦ T
1
T
1
: R
2
→ R
2
, T
1
(x, y) = (x − 2y, ,2x + 3y)
T
2
: R
2
→ R
2
, T
2
(x, y) = (2x, x − y);
T
1
: R
2
→ R
3
, T
1
(x, y) = (x − 2y, x y, x ,+ − y)
T
2
: R
3
→ R
2
, T
2
(x, y, z) = (x − 3y, 3x + z);
T
1
: R
3
→ R
2
, T
1
(x, y, z) = (x − 3y, ,3x + z)
T
2
: R
2
→ R
3
, T
2
(x, y) = (x − 2y, x y, x .+ − y)
T
: R R
3
→
3
, T (x, y, z) = (x, x + y, x + y + z)
T
: R R
3
→
3
, T (x, y, z) = (x + y, y z, x+ + z)
T
: R
4
→ R
4
, T (x
1
, x , x , x , x , x , x
2 3 4
) = (x x
1
− 2
2 2 3
+ x
4 3
)
T
: R
4
→ R
4
, T (x
1
, x , x , x , x , x , x
2 3 4
) = (x
4 3 2 1
)
A T B, B
′
[v]
B
[T (v)]
B
′
T (v) B
′
[T (v)]
B
′
= A[v]
B
T
: R
2
→ R
2
, T (x, y) = (2x − 12y, x − 5y), v ,= (10, 5)
B = B
′
= {(4, ,1) (3, 1) ;}
T
: R R
2
→
3
, T (x, y) = (x + y, x, y), v ,= (5, 4)
B = {(1, , , ,−1) (0, , ,1)}, B
′
= {(1 1 0) (0, 1, 1) (1, 0, 1) ;}
T
: R R
3
→
2
, T (x, y, z) = (x − y, y −z), v ,= (1, 2, −3)
B = {(1, 1, , , ,1), ,(1 1 0), , , ,(0 1 1)}, B
′
= {(1 2) (1 1) ;}
T
: R
3
→ R
3
, T (x, y, z) = (x + y + z, −x + 2z, 2y −z), v = (4, −5, 10),
B = {(2, ,0, , ,1), ,(0 2 1), , , ,(1 2 1)}, B
′
= {(1 1 1), , ,(1 1 0) (0, 1, 1)}.
T
: R
3
→ R
3
T (x, y, z) = (x + y + z, −x + 2y + 3z, .2x − y + z)
T
R
3
T { }(1, 2, −1) (1 0) (0 0), , 0, , , 1,
R
3
A
′
T B
′
A
′
A T B P B
′
B
P
−1
A AP
′
= P
−1
T
: R
2
→ R
2
, T (x, y) = (2x − y, −x + y),
B
′
= {(1, , ,−2) (0 3) ;}
T
: R
3
→ R
3
, T (x, y, z x, y, z ,) = ( )
B
′
= {(1, , , , , ,1 0), (1 0 1) (0, 1 1) ;}
T
: R
3
→ R
3
, T (x, y, z) = (x, x + 2y, x ,+ y + 3z)
B
′
= {(1, , , , , ,−1 0) (0 0, 1) (0 1, −1)}.
B = {(1, 3), ,(−2 −2)} B
′
= {(−12, , ,0) (−4 4)}
R
2
A
=
3 2
0 4
T
: R R
2
→
2
B
P B
′
B
A P [v]
B
[T (v)]
B
[ ]
v
B
′
=
−1
2
.
[ [ ]v]
B
= P v
B
′
, [T (v)]
B
= A[v]
B
= AP [v]
B
′
A
′
T
: R
2
→ R
2
B
′
P
−1
[T (v)]
B
′
[T (v)]
B
′
= A
′
[ ]v
B
′
[
T (v)]
B
′
= P
−1
[T (v)]
B
2 B = {(1, , , ,1) (−2, 3)}, B
′
= {(1, −1) (0 1)}
A
=
3 2
0 4
,
[
v]
B
′
=
1
−
3
.
2 B = {(1, , , ,2) (−1 −1)}, B
′
= {(−4, 1), (0 2)}
A
=
2 1
0
−1
,
[ ]
v
B
′
=
−1
4
.
2 B = {(1, ,−1), , ,(−2 1)}, B
′
= {(−1, 1) (1 2)}
A
=
2 1
0
−1
,
[ ]
v
B
′
=
1
−
4
.
2 B = {(1, , , , , , , ,1, ,0) (1 0 1) (0, 1, 1)}, B
′
= {(1 0, 0) (0, 1, 0) (0, 0 1)}
A
=
3
2
−1 −
1
2
−
1
2
2
1
2
1
2
1
5
2
,
[
v]
B
′
=
1
0
−1
.
2 B = {(1, , ,0, , , ,0), , ,(0 1 0) (0 0, 1)}, B
′
= {(1 1 −1) (1, , ,−1, 1) (−1 1, 1)}
A
=
3
2
−1 −
1
2
−
1
2
2
1
2
1
2
1
5
2
,
[
v]
B
′
=
2
1
1
.
λ
i
x
i
A
=
1 0
0
−1
,
λ
1
= 1, x ,
1
= (1, 0)
λ
2
= −1, x
2
= (0, 1);
A
=
2 3 1
0 −1 2
0 0 3
,
λ
1
= 2, x ,
1
= (1, ,0 0)
λ
2
= −1, x ,
2
= (1, −1, 0)
λ
3
= 3, x .
3
= (5, 1, 2)
x
i
A
A
=
7 2
2 4
, x
1
= (1, 2) 1) 2) 0);, x
2
= (2, , x
3
= (1, − , x
4
= (−1,
A
=
−1 −1 1
−2 0 −2
3 −3 1
, x
1
= (2 = (2 = (2 = (, ,−4 6), x
2
, 0, 6), x
3
, 2, 0), x
4
−1 0, , 1).
A
=
6 −3
−
2 1
A
=
7 2
2 4
A
=
2 0 1
0 3 4
0 0 1
A
=
−5 0 0
3 7 0
4 −2 3
A
=
2 −2 3
0 3 2−
0 −1 2
A
=
3 2 1
0 0 2
0 2 0
A
=
1 2 2−
−2 5 −2
−6 6 −3
A
=
3 2 3−
−3 −4 9
−1 −2 5
A
P
−1
AP
A
=
1 3
−
1 5
, P =
3 1
1 1
A
=
−1 1 0
0 3 0
4 −2 5
, P =
0 1 3−
0 4 0
1 2 2
A
=
0 0
2 0
A
=
1
1
2
− −
2 1
A
=
1 −2 1
0 1 4
0 0 2
A
=
2 1 1−
0 −1 2
0 0 1−
A P A
P
−1
AP
P
−1
AP
A
=
6 −3
−
2 1
A
=
7 2
2 4
A
=
2 0 1
0 3 4
0 0 1
A
=
−5 0 0
3 7 0
4 −2 3
A
=
2 −2 3
0 3 2−
0 −1 2
A
=
3 2 1
0 0 2
0 2 0
A
=
1 2 2−
− −2 5 2
− −6 6 3
A
=
3 2 3−
−3 −4 9
−1 −2 5
B T B
T
: R R
2
→
2
, T (x, y) = (x + y, x + y)
T
: R
3
→ R
3
, T (x, y, z) = (−2x + 2y −3z, 2x + y − 6z, −x − 2y)
A
=
1 0 0
0 2 0
0 0 3
, B =
3 0 0
0 2 0
0 0 1
.
A B
P
B
= P
−1
AP.
A
=
−1 a −1
−3 5 −1
−3 3 1
a
a 2 A
a = 3 P
P
−1
AP
A
=
6 −2
−
2 1
;
A
=
1 3
2 4
;
A
=
2 −2 1
−2 3 4
0 4 1
;
A
=
−5 3 4
3 7 2−
4 −2 3
.
A
=
1 3
3 1
A
=
0 2
2 0
A
=
2 1 1
1 2 1
1 1 2
A
=
0 2 2
2 0 2
2 2 0
A
=
0 4 4
4 2 0
4 0 2−
A
=
0 1 1
1 0 1
1 1 0
A
=
2 −1 −1
−1 2 −1
−1 −1 2
A
=
3 0 0
0 1 0
0 0 1
A
=
√
2
2
√
2
2
−
√
2
2
√
2
2
A
=
2 2/3 − /3
2
/3 1/3
A
=
−4 0 3
0 1 0
3 0 4
A
=
−4/5 0 3 5/
0 1 0
3/5 0 4 5/
P
P
T
AP
P
T
AP
A
=
1 1
1 1
A
=
4 2
2 4
A
=
0 3 0
3 0 4
0 4 0
A
=
1 −1 2
−1 1 2
2 2 2
A
=
1 1 0 0
1 1 0 0
0 0 1 1
0 0 1 1
A
=
1 a 0
a 1 0
0 0 3−
a
A a A 3
a = 2 P
P
t
AP
x
2
+ y
2
9
x
2
+ 10y
2
− 5xy
x
2
+ 2y
2
+ 3z
2
− 2xy −4xz − 5yz
x x
2
+ 2
2
+ 3z
2
+ 4t
2
+ xy −2xz + 6xt − 4yz − 8yt
A P
P
T
AP
A
2 2 3
x
2
− y
2
− xy
5 2
x
2
+ 5y
2
− xy
16
x
2
+ 9 + 24y
2
xy
xy, xz, yz
x
2
+ y
2
+ 4xy −9 = 0
2 4
x
2
+ 5y
2
− xy −36 = 0
xy + x −2y + 3 = 0
5 2
x
2
+ 5y
2
− xy + 10
√
2x = 0
3 2
x
2
+ 3 + 8y
2
z
2
− xy −16 = 0
2
x
2
+ 2 + 2 + 2 + 2y
2
z
2
xy xz + 2yz − 1 = 0
x
2
+ y
2
+ z
2
+ 2xy −z − 8 = 0

Tài liệu khác của Trường Đại học Công nghệ, Đại học Quốc gia Hà Nội

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2x − 5y = 10 x + y + z = 1  x  1 − x2 = 2 x − y = 4  x2 = 3 2y + z = 6  3z = 6  −x + y − z = 0  2y + z = 3  x1 + x2 + x3 + x4 = 10  10z = 0 x2 + 3x3 + 4x4 = 15  x + 2y = 7  3x − 2y + 4z = 1  2x + y = 8 x + y − 2z = 3  2x − 3y + 6z = 8  x + 2y = 0   5x − 3y + 2z = 3 x + y = 6  2x + 4y − z = 7  3x − 2y = 8  x − 11y + 4z = 3   x + y + z = 6 x + y + z + w = 6     2x − y + z = 3 2x + 3y − w = 0 −3x + 4y + z + 2w = 4  3x − z = 0    x + 2y − z + w = 0  x + y + z = 2  4x + 3y + 17z = 0   −x + 3y + 2z = 8 5x + 4y + 22z = 0  4x + y = 4  4x + 2y + 19z = 0  2 3  2 1 3  + = 0 + − = 4  x y    x y z 3 4 25   4 2  = + = 10  − − x y 6 x z   2 3 13    − + − = −8 x y z k  4x + ky = 6  x + y + kz = 3  kx + y = −3 x + ky + z = 2  kx + y + z = 1  x + 2y + kz = 6 3x + 6y + 8z = 4  −1 3   1 2 3   1 2 3 4 5  0 −2 0 1 6     0 1 9 1 0 0 0 0  0 1 0 0 0   1 2 3 4 5   0 0 1 0 1     1 2 3   0 4 3 2 1  0 0 0 9 9    0 1 1   0 0 1 1 1  0 0 0 0 0 0 9 9  1 0 0   1 2 1 0   2 1 1 0  0 1 2 0 0 1 −1 1    −2 1 −2  0 0 0 0 1 0 1 0  1 2 0 1 4   1 −1 0 3   2 1 −1 3   0 −1 −2 −1 −3     0 1 −2 1   1 −1 1 0   0 0 1 2 1  0 0 1 −1 0 1 2 1 0 0 0 −1 −4  −3x + 5y = −22  2x − y + 3z = 24   3x + 4y = 4 2y − z = 14  4x − 8y = 32  7x − 5y = 6  x − 3z = −2  x + 2y + z = 8  3x + y − 2z = 5 −3x − 6y − 3z = −21  2x + 2y + z = 4  2x + y − z + 2w = −6   x + y − 5z = 3   3x + 4y + w = 1  x − 2z = 1 x + 5y + 2z + 6w = −3    2x − y − z = 0  5x + 2y − z − w = 3  1 k 2  A := .  −3 4 1  −3 6 −6 A k A k  2 −1 3  B := .  −4 2 k  4 −2 6 B k B k  1 0 1 1   −1 2 1   −1 1 2 1   2 3 1 10  0 1 2 1 0 1 0 2 3 1 −2 2   −3 −3 22       0 0 0 1 0 0 1 5 4 2 4 4 −2 3 −2  2x + 3z = 3
 4x + 12y − 7z − 20w = 22  4x − 3y + 7z = 5 3x + 9y − 5z − 28w = 30  8x − 9y + 15z = 10  x + 2y + 6z = 1   2x + 3y + 3z = 3 2x + 5y + 15z = 4  6x + 6y + 12z = 13  3x + y + 3z = −6  12x + 9y − z = 2  2x + y + 2z = 4  2x + 2y = 5  2x + 6z = −9   2x − y + 6z = 2 3x − 2y + 11z = −16   3x − y + 7z = −11 2x + y + z + 2w = −1    5x − 2y + z − 3w = 0  2x + 5y − 19z = 34 −x + 3y + 2z + 2w = 1  3x + 8y − 31z = 54   3x + 2y + 3z − 5w = 12 k  x − y + 2z = 0  −x + y − z = 0  x + ky + z = 0 a, b, c  x + 2y = 3  x + y = 0  ax + by = −9   y + z = 0 x + z = 0    ax + by + cz = 0  x + y = 2     y + z = 2 2x − y + z = a  x + z = 2 x + y + 2z = b    ax + by + cz = 0  3y + 3z = c A + B, A − B, 2A, B + 1 A A, B 2  6 −1   1 4   3 2 −1   0 2 1  A = 2 4 A = 2 4 5 5 4 2  , B = , B =   −1 5      −3 5 1 10 0 1 2 2 1 0  2 3 4   0 6 2   2 1 1   2 −3 4  A = 0 1 −1 , B = 4 1 0 A = , B =     −1 −1 4 −3 1 −2 2 0 1 −1 2 4 ABt, BAt, AtB A, B  −4 0   1 2  X A = 1  −5 , B =   −2 1  −3 2 4 4 AC = BC A = B  0 1   1 0   2 3  A = , B = , C = 0 1 1 0 2 3  1 2 3   4 −6 3   0 0 0  A = 0 5 4 , B = 5 4 4 , C = 0 0 0       3 −2 1 −1 0 1 4 −2 3  x y   y z   4 x   w x   −4 3   y w  4 = 2 + 2 = + 2 z −1 −x 1 5 −x y x 2 −1 z x  1 2   1 0   1 2   6 3  A = A = 3 5 0 1 3 4 19 2  2 −1   1 0   2 1   3 17  A = A = 3 −2 0 1 3 1 4 −1  x y   1 1  AB = BA A = , B = z w −1 1 b A A, b  1 −1 2   −1   1 1 −5   3  A = , b = 3 −3 1 7 A = 1 0 −1 1  , b =    2 −1 −1 0  1 2 4   1   −3 5   −22  A =  −1 0 2 , b = , b =   3  A = 3 4 4     0 1 3 2 4 −8 32  1   1   2   1   0  X =  0 , Y = , Z = , W = , O =   1   −1   1   0  1 0 3 1 0 a b Z = aX + bY a b W = aX + bY aX + bY + cW = O a = b = c = 0 a, b c aX + bY + cZ = O  1 0 0  A5 A10 A = .  0 −1 0  0 0 2 B A A, B  1 2   −2 1   1 −1   3 1  A = , B = A = , B = 1 3 4 3 −1 2 3 5 −2 1 2 2  −2 2 3   −4 −5 3   2 −17 11   1 1 2  A = 1 4 8 3 1 11 7 , B = 2 4 −3  −1 0  , B = 1 − − A = − − 3       0 1 4 1 2 0 0 3 −2 3 6 −5  1 2   1 2 2   10 5 −7   3 2 5  3 7 3 7 9 2 2 4    −5 1 4    1 4 3 2 −2  − − −7 −4 4 0 sin θ cos θ  − cos θ sin θ  1 1 1   1 2 −1   1 1 2   0.1 0.2 0.3  3 5 4 3 7 3 1 0 . .    −10     −0.3 0 2 0 2  3 6 5 7 16 −21 −2 0 3 0.5 0.5 0.5
(AB)−1, (At)−1, A−2, (2B)−1  2 5   7   −2 1   5 2  A −3 −1 = , B−1 = A−1 = 7 7 , B−1 = 11 11 −7 6 2 0 3 2 3 − 1 7 7 11 11  1 −1 3   2 4 5   1 −4 2   6 5 −3  2 4 2 A−1 = 3 1 2 1 A−1 = 0 1 3  −2 , B−1 = −3 , B−1 = −2 4 −1 2 2   4 4      1 1 1 1 1 2 4 2 1 1 3 4 4 2 4 2 x A = A−1  3 x   2 x  A = A = −2 −3 −1 −2 x A A A  1 2   2 4  (2A)−1 = (4A)−1 = 3 4 −3 2 Ax = b A−1  5x + 4y = 2  3x + 2y = 1 −x + y = −22 x + 4y = −3  −x + y + 2z = 1  x + y + 2z = 0   2x + 3y + z = −2 x − y + z = −1  5x + 4y + 2z = 4  2x + y + z = 2  1 2   λ − 3 2   1 2 −3   2 4 −3  3 4 4 λ − 1 6 5 4 1 3 6     1 −3 2 −8 −7 4  1 0 2   x y 1   1 4 3 2   3 0 7 0  2 3 1 2 6 11 12  −1 1 4     −5 6 2 1    2 0 3 0 −1 1  0 0 0 0   4 1     −1 2  3 −2 1 5 1 5 2 10    x y 1 1 4 −2   −2 −2 1   3 2 0  1 5 1 −1 4 3  5 2 0 0 −2   2 6 6 2   5 3 0 6   0 1 4 3 2   2 −1 3     2 7 3 6   4 6 4 12   0 0 2 6 3  1 4 4  1 5 0 1   0 2   0 0 3 4 1       −3 4    1 0 2 3 7 0 7 0 1 −2 2 0 0 0 0 2 x          x + 3 1   x − 2 −1   x 2 0   x 0 1   = 0 = 0 2 x + 2           −3 x   0 x + 1 2  = 0  0 x 3  = 0  0 1 x   2 2 x − 2           1 1 1   a + b a a   a b c   a a + b a  
 = (a − b)(b − c)(c − a)   = b2(3a + b)  a2 b2 c2   a a a + b           a 1 1 1   1 + a 1 1   1 1 1     1 1 + b 1  = abc 1 + + +  1 a 1 1      = (a + 3)(a − 1)3  c 1 1 1 + c  a b  1 1 a 1       1 1 1 a  (a, b, c 6= 0)  1 7 −3   3 −1 −3   4 −7 9 1   0 −3 8 2  1 3 1 4 2 6 2 7 0 8 1    −1 − −     −1 6  4 8 1 3 −1 −1  3 6     −3 3   −4 6 0 9  0 7 4 −1 −7 0 0 14  9 −4 2 5   1 1 1   4 5 −2   2 7 6 −5  2 3 4 3  4 1   −1 −2     −2 0  1 −2 −1 −2 1 4 7 3 4 10 |A|, |B|, AB, |AB| |A||B| = |AB|  −1 2 1   −1 1 0   2 0 1 1   1 0 −1 1  A = 1 0 1 , B = 0 2 0 1 2 1 0 2     A −1 0 1 =   , B =   0 1 0 0 0 3      2 3 1 0   1 1 −1 0  1 2 3 0 3 2 1 0  3 2 4 0   4 2 −1 0   2 0 1   2 −1 4  1 −1 2 1 1 1 2 −1 A =   , B =   A = 1 0 1 3  0 0 3 1   0 0 2 1   −1 2 , B =        3 1 0 3 −2 1 −1 1 1 0 −1 0 0 0
|At|, |A2|, |AAt|, |2A|, |A−1|  2 0 5   1 5 4  A =  4 −1 6  A =  0 −6 2  3 2 1 0 0 −3  1 2   1 0 0   −3 −5 −7  3 4 0 2 6 2 4 3     0 −4 −12 0 1 −1  1 2 3   0 1 1   −1 0  0 1 −1 1 2 3     0 4 2 2 2 −1 −1 −2  x + 2y = 5  3x + 3y + 5z = 1  −x + y = 1 3x + 5y + 9z = 2  5x + 9y + 17z = 4  4x − y − z = 1   2x + 3y + 5z = 4 2x + 2y + 3z = 10  3x + 5y + 9z = 7  5x − 2y − 2z = −1  5x + 9y + 17z = 13  4x − 2y + 3z = −2  2x + y + 2z = 6   2x + 2y + 5z = 16 −x + 2y − 3z = 0  8x − 5y − 2z = 4  3x + 2y − z = 6 k k  kx + (1 − k)y = 1 (1 − k)x + ky = 3 (0, 0), (2, 0), (0, 3); (1, 1), (2, 4), (4, 2). (1, 2), (3, 4), (5, 6); (−1, 0), (1, 1), (3, 3). (−4, 7), (2, 4); (−2, 3), (−2, −4).
(1, 1, 1), (0, 0, 0), (2, 1, −1), (−1, 1, 2);
(3, −1, 1), (4, −4, 4), (1, 1, 1), (0, 0, 1).
(−4, 1, 0), (0, 1, 2), (4, 3, −1), (0, 0, 1);
(0, 0, −1), (0, −1, 0), (1, 1, 0), (2, 1, 2);
(1, 2, 3), (−1, 0, 1), (0, −2, −5), (2, 6, 11);
(1, 2, 7), (−3, 6, 6), (4, 4, 2), (3, 3, 4).
(1, −2, 1), (−1, −1, 7), (2, −1, 3);
(0, 0, 0), (1, −1, 0), (0, 1, −1);
(0, −1, 0), (1, 1, 0), (2, 1, 2);
(1, 2, 7), (4, 4, 2), (3, 3, 4). = 0 A = 1 B = 2 C = 3 D = 4 E = 5 F = 6 G = 7 H = 8 I = 9
J = 10 K = 11 L = 12 M = 13 N = 14 O = 15 P = 16 Q = 17 R = 18 S = 19 T = 20 U = 21 V = 22 W = 23 X = 24 Y = 25 Z = 26.  1 1 1  A = .  0 1 1  1 1 2 D T H U B A  20 8 21  D = . 20 8 21 0 2 1, 0 2 1  1 1 1   20 8 21   41 49 70  DA = 0 1 1 = 0 2 1   1 3 4 1 1 2  27 42 56  A B = 29 30 55 A  1 1 1   33 51 60  A = 1 2 2 . B =   30 31 56 1 1 2 A
u = (1, 2, 3), v = (2, 2, −1), w = (4, 0, −4).
v−u, u−v+2w, 2u+4v−w, 5u−3v− 1w. z 2u + v − w + 3z = 0. 2 z 2z − 3u = w. R4 M1,4 P3 P3 M2,3 M2,2 3. 1 f (x) = ax + b  a b  ; c 0 1 f (x) = ax a 6= 0  a b  ; {(x, y) ∈ R2 : x > 0} c 1
{(x, y) ∈ R2 : x > 0, y > 0} 2 × 2 0 {(x, y) ∈ R2 : x = 2y} 2 × 2 0 R2
(x1, y1) ⊎ (x2, y2) = (x1 + x2, y1 + y2) c ∗ (x, y) = (cx, y) √ √
(x1, y1) ⊎ (x2, y2) = (x1 + x2, y1 + y2) c ∗ (x, y) = ( cx, cy)
(x1, y1) ⊎ (x2, y2) = (x1, 0) c ∗ (x, y) = (cx, cy)
(x1, y1) ⊎ (x2, y2) = (x1x2, y1y2) c ∗ (x, y) = (cx, cy) R3
(x1, y1, z1) ⊎ (x2, y2, z2) = (0, 0, 0) c ∗ (x, y, z) = (cx, cy, cz)
(x1, y1, z1) ⊎ (x2, y2, z2) = (x1 + x2 + 1, y1 + y2 + 1, z1 + z2 + 1) c ∗ (x, y, z) = (cx, cy, cz) W V
W = {(x, y, z, w) ∈ V = R4 : w = 0}
W = {(x, y, z) ∈ V = R3 : x, y, z ∈ Q}
W = {(x, y, z) ∈ V = R3 : z = 2x + 3y}
W = {(x, y) ∈ V = R2 : y = x2}  0 b   W = ∈ V = M W = = 0 a
{(x, y, z) ∈ V = R3 : z = 1/x, x 6 } 0 2,2 : a, b ∈ R  a b  
W = {(x, y, z) ∈ V = R3 : x2 + y2 = z2}   W = a + b 0 V = M R   ∈ 3,2 : a, b, c ∈ W = {x ∈ V = Rn : Ax = 0} A  0 c  m × n
W = {(x, y, z) ∈ V = R3 : z = 1}
W = {x ∈ V = Rn : Ax = b 6= 0} A
W = {(x, y, z) ∈ V = R3 : x, y, z ∈ Z} m × n u, v S
S = {(2, −1, 3), (5, 0, 4)}, u = (1, 1, −1), v = (−1, −2, 2), w = (1, −8, 12)
S = {(1, 2, −2), (2, −1, 1)}, u = (1, −5, −5), v = (−4, −3, 3), w = (−2, −6, 6) u = (1, 2), v = (1, −1). u v (2, 1) (3, 0) (0, 3) (1, −1) z u, v w z = (10, 1, 4)
u = (2, 3, 5), v = (1, 2, 4), w = (−2, 2, 3) z = (−1, 7, 2)
u = (1, 3, 5), v = (2, −1, 3), w = (−3, 2, −4) z = (0, 5, 3, 0)
u = (1, 1, 2, 2), v = (2, 3, 5, 6), w = (−3, 1, −4, 2) z = (2, 5, −4, 0)
u = (1, 3, 2, 1), v = (2, −2, −5, 4), w = (2, −1, 3, 6) {(0, 0), (1, 2)}
{(1, 1, 1), (2, 2, 2), (1, 2, 3)} {(1, 0), (1, 1), (2, −1)}
{(−4, −3, 4), (1, −2, 3), (6, 0, 0)} {(1, −4, 1), (6, 3, 2)}
{(1, 0, 0), (0, 4, 0), (0, 0, −6), (1, 5, −3)} {(6, 2, 1), (−1, 3, 2)}
{(0, 0, 0, 1), (0, 0, 1, 1), (0, 1, 1, 1), (1, 1, 1, 1)} 0 {(3, 4), (−1, 1), (2, 0)}
{(1, 1, 1), (1, 1, 0), (0, 1, 1), (0, 0, 1)} {(2, 4), (−1, −2), (0, 6)}
{(1, 2, 3, 4), (1, 0, 1, 2), (1, 4, 5, 6)} t
{(t, 1, 1), (1, t, 1), (1, 1, t))}
{(t, 0, 0), (0, 1, 0), (0, 0, 1))}
{(t, 1, 1), (1, 0, 1), (1, 1, 3t))}
{(t, t, t), (t, 1, 0), (t, 0, 1))}  2 −3   0 5  A = , B = . 4 1 1 −2 A B  6 2   0 0   6 −19   −2 28  9 11 0 0 10 7 1 −11
 1 −1   4 3   1 −8  , , 4 5 −2 3 22 23 P2
{2 − x, 2x − x2, 6 − 5x + x2}
{x2 + 3x + 1, 2x2 + x − 1, 4x} {x2 − 1, 2x + 5} {x2, x2 + 1} R2 {(2, 1), (−1, 2)} {(1, −2), (−3, 9)} {(−1, 4), (4, −1), (1, 1)} {(1, 1)} {(1, 3), (−2, −6), (3, 9)} {(1, −2), (−2, 1), (1, 1)} R3
{(4, 7, 3), (−1, 2, 6), (2, −3, 5)}
{(1, 1, 0), (1, 0, 1), (0, 1, 1)}
{(6, 7, 6), (3, 2, −4), (1, −3, 2)}
{(1, −2, 0), (0, 0, 1), (−1, 2, 0)} {(−2, 5, 0), (4, 6, 3)}
{(1, 0, 3), (2, 0, −1), (4, 0, 5), (2, 0, 6)} {1, x2, x2 + 2} P2
{x2 − 2x, x3 + 8, x3 − x2, x2 − 4} P3 R4 M2,3 M4,1 M2,2 R2 {(1, 2), (1, 0), (0, 1)} {(−4, 5), (0, 0)} {(1, 2), (3, 4)} {(1, 1)} {(1, −2), (−1, 2), (2, 4)} R3
{(1, 3, 0), (4, 1, 2), (−2, 5, −2)} {(1, 1, 1), (1, 2, 3)}
{(1, 5, 3), (0, 1, 2), (0, 0, 6)} P2 {1, 2x, x2 − 4, 5x}
{6x − 3, 3x2, 1 − 2x − x2}
{1 − x, 1 − x2, 3x2 − 2x − 1}
{x − 1, x2 − 1, 1 − 2x − x2} R3 u = (8, 3, 8)
{(4, 3, 2), (0, 3, 2), (0, 0, 2)}
{(1, 1, 1), (1, 1, 0), (1, 0, 0)}
{(1, 4, 7), (3, 0, 1), (2, 1, 2)} M2,2
 2 0   1 4   0 1   0 1  , , , 0 3 0 1 3 2 2 0 
1 2   2 −7   4 −9   3 −7  , , , −5 4 6 2 10 12 1 2 2 3 R2 {(1, 0), (0, 1), (1, 2)} R3
{(1, 3, −2), (−4, 1, 1), (−2, 7, −3), (2, 1, 1)} R2 {(1, 1)} R3 {(1, 0, 2), (0, 1, 1)} R2 W = {(2t, t) : t ∈ R} W = {(0, t) : t ∈ R} R3 W = {(2t, t, −t) : t ∈ R}
W = {(2s − t, s, t) : t, s ∈ R} R4
W = {(2s − t, s, t, s) : t, s ∈ R}
W = {(t, 2s − 3t, w, t) : t, s, w ∈ R}
W = {(5t, −3t, t, t) : t ∈ R} 1 1  2 −3 1   1 2 3   2 4 −3 −6  5 10 6 7 14 −6 −3     8 −7 5  2 4 1   − − −2 −2 −4 4 5    2 4 −2 −2  1 2 3  3 6 −6 −4   1 −3 2 −2 −4 4 9 R3 S S
{(1, 2, 4), (−1, 3, 4), (2, 3, 1)}
{(1, 1, 2), (4, 4, 8), (1, 1, 1)}
{(4, 2, −1), (1, 2, −8), (0, 1, 2)}
{(1, 2, 2), (−1, 0, 0), (1, 1, 1)} R4 S S
{(2, 9, −2, 53), (−3, 2, 3, −2), (8, −3, −8, 17), (0, −3, 0, 15)}
{(2, 5, −3, −2), (−2, −3, 2, −5), (1, 3, −2, 2), (−1, −5, 3, 5)} Ax = 0 A  1 4 2   1 2 3   1 2 −3   1 3 −2 4  1 0 0 2 −1 4 0 1 −1 2  2 −1      4 3 −2 −2 −6 4 −8 1 3  −x + y + z = 0  x − 2y + 3z = 0  −3x − y = 0 −3x + 6y − 9z = 0  2x − 4y − 5z = 0  x + 2y − 4z = 0 −3x − 6y + 12z = 0  4x − y + 2z = 0  3x + 3y + 15z + 11t = 0   2x + 3y − z = 0 x − 3y + z + t = 0  3x + y + z = 0  2x + 3y + 11z + 8t = 0 Ax = b x = xh + xp xh Ax = 0 xp Ax = b  x + 3y + 10z = 18  3x − 6y + z = 12     −2x + 7y + 32z = 29 −7x + 14y + 4z = −28 −x + 3y + 14z = 12  2x − 4y + 5z = 8    x + y + 2z = 8 b A b A  −1 2   3   −1 2   2   1 3 0   1  A = , b = A = , b = 4 0 4 2 −4 4 A = 2  −1 1 0 , b =    0 1 1 −3 x x B
B = {(2, −1), (0, 1)}, [x]B = [4, 1]t
B = {(1, 0, 1), (1, 1, 0), (0, 1, 1)}, [x]B = [2, 3, 1]t
B = {(0, 0, 0, 1), (0, 0, 1, 1), (0, 1, 1, 1), (1, 1, 1, 1)}, [x]B = [1, −2, 3, −1]t x B
B = {(−6, 7), (4, −3)}, x = (−26, 32)
B = {(8, 11, 0), (7, 0, 10), (1, 4, 6)}, x = (3, 19, 2)
B = {(9, −3, 15, 4), (3, 0, 0, 1), (0, −5, 6, 8), (3, −4, 2, −3)}, x = (0, −20, 7, 15) B B′
B = {(1, 0), (0, 1)}, B′ = {(2, 4), (1, 3)}
B = {(1, 1), (1, 0)}, B′ = {(1, 0), (0, 1)}
B = {(1, 0, 0), (0, 1, 0), (0, 0, 1)}, B′ = {(1, 0, 0), (0, 2, 8), (6, 0, 12)}
B = {(1, 0, 0), (0, 1, 0), (0, 0, 1)}, B′ = {(1, 3, −1), (2, 7, −4), (2, 9, −7)} P B B′ Q B′ B P = Q−1 [x]B
B = {(1, 3), (−2, −2)}, B′ = {(−12, 0), (−4, 4)}, [x]B′ = [−1, 3]t
B = {(2, −2), (6, 3)}, B′ = {(1, 1), (32, 31)}, [x]B′ = [2, −1]t
B = {(1, 0, 2), (0, 1, 3), (1, 1, 1)}, B′ = {(2, 1, 1), (1, 0, 0), (0, 2, 1)}, [x]B′ = [1, 2, −1]t
B = {(1, 1, 1), (1, −1, 1), (0, 0, 1)}, B′ = {(2, 2, 0), (0, 1, 1), (1, 0, 1)}, [x]B′ = [2, 3, 1]t P2 : p = x2 + 11x + 4 p = 3x2 + 114x + 13 p = −2x2 + 5x + 1 p = 4x2 − 3x − 2 X M3,1  0   1  X =  3  X =  2  2 −1  2   1  X = X = 0  −1    4 −4 Rn v = (2, 0, −4, 5) v = (1, −3, −5, 6, 2) v = (8, 8, 6) u u v u v u = (1, 2, −4, 3) v = (5, 1, 2, 3) u = (1, −3, −5, 4, 2) u = (2, −1, −2, 3, 1) u = (0, 1, 2, 3) v = (1, 0, 4, −1) kuk2 = 4, kvk2 = 10 hu, vi = −5 hu + v, 2u − vi kuk2 = 8, kvk2 = 6 hu, vi = 7 h3u − v, u − 3vi u = (1, 1, −2) v = (1, −3, −2) u = (1, 2, 3, 4) v = (4, 3, 2, 1) u = (1, 1, 1) v = (0, 1, −2) u = (−1, 1, −1, 1) v = (1, 2, 3, 4) u = (3, 1) v = (−2, 4) u = (1, 0, 1, 0) v = (3, 3, 3, 3)  π π   3π 3π  u = cos , sin v = cos , sin 6 6 4 4  π π   π π u = cos , sin v = cos , sin 3 3 4 4 u = (2, 7) u = (2, −1, 1) u = (0, 0, −1, 1) u = (cos x, sin x, −1) v = (sin x, − cos x, 0) u = (− sin x, cos x, 1) v = (sin x, − cos x, 0) hu, vi, kuk, kvk, d(u, v) u = (4, 3) v = (0, 5) hu, vi = 3u1v1 + u2v2 u = (1, 1, 1) v = (2, 5, 2) hu, vi = u1v1 + 2u2v2 + 3u3v3 hA, Bi, kAk, kBk, d(A, B)
hA, Bi = 2a11b11+a12b12+a21b21+2a22b22  −1 3   0  A = −2 , B = ; 4 −2 1 1  1 −1   0 1  A = , B = . 2 4 −2 0 hu, vi R2 hu, vi = u1v1 hu, vi = u1v1 − u2v2 hu, vi = u2v2 + u2v2 1 1 2 2 hu, vi = u1u2 + v1v2  0 3    A = −3 1 , B = a 2 1 4 3
hA, Bi = 2a11b11 + a12b12 + a21b21 + 2 22b22;  0 1   1 1  A = , B = hA, Bi = a a a22b22. 2 −1 2 −2 11 b11 + 2 12b12 + a21b21 + 2 u = (−4, 3) v = (0, 5) hu, vi = 3u1v1 + u2v2 u = (1, 1, 1) v = (2, −2, 2) hu, vi = u1v1 + 2u2v2 + u3v3 proj v proj u u v u = (1, 2) v = (2, 1) proj v proj u R2 u v u = (−1, 3) v = (4, 4) proj v proj u R2 u v u = (1, 3, −2) v = (0, −1, 1) u = (0, 1, 3, −6) v = (−1, 1, 2, 2) u = (5, 1, 4) v = (2, 1, −1) u v hu, vi = u1v1 + 2u2v2 + 3u3v3 u v w = (2, 7) hu, vi = u1v1 + 3u2v2 w = (2, −1, 1) hu, vi = 2u1v1 + 3u2v2 + u3v3 w = (−1, 1, −1, 1)
hu, vi = u1v1 + 3u2v2 + 3u3v3 + u4v4
S = {(4, −1, 1), (−1, 0, 4), (−4, −17, −1)} √ √ √ √ √ √ √ √ n 2 2  6 6 6  3 3 3 o S = , 0, , − , , , , , − 2 2 6 3 6 3 3 3 √ √ √ √ n 2 2  2 2   1 1 1 1o S = , 0, 0, , 0, , , 0 , − , , − , 2 2 2 2 2 2 2 2 x √ √ √ √ n 5 2 5  2 5 5 o S = , , − , x = (−3, 4) 5 5 5 5 n 3 4   4 3  o S = , , 0 , − , , 0 , (0, 0, 1) x = (5, 10, 15) 5 5 5 5
S = {(1, −2, 2), (2, 2, 1), (2, −1, −2)}
S = {(4, −3, 0), (1, 2, 0), (0, 0, 4)}
S = {(1, 0, 0), (1, 1, 1), (1, 1, −1)}
S = {(0, 1, 2), (2, 0, 0), (1, 1, 1)}
S = {(0, 1, 1), (1, 1, 0), (1, 0, 1)}  x  1 + x2 − 3x3 − 2x4 = 0 x1 + x2 − x3 − x4 = 0  2x1 − x2 − 2x4 = 0 2x1 + x2 − 2x3 − 2x4 = 0;  3x1 + x2 − 5x3 − 4x4 = 0;  x1 − x2 + x3 + x4 = 0  x1 + x2 − 3x3 + 2x4 = 0 x1 − 2x2 + x3 + x  4 = 0; x1 + 2x2 − 3x3 + 4x4 = 0  2x1 + x2 − 6x3 + 2x4 = 0; x1 − 2x2 + x3 = 0.
V1 = span{(2, 1, −1), (0, 1, 1)} V2 = span{(−1, 2, 0)}
V1 = span{(0, 0, 2, 1), (0, 0, 1, −2)}
V2 = span{(3, 2, 0, 0), (0, 1, −2, 0)} V = span{(1, 2, 3), (1, 1, 1)}
V = span{(1, 2, 0, 0), (0, 1, 0, 1)} v V
V = span{(0, 0, −1, 1), (0, 1, 1, 1)} v = (1, 0, 1, 1) V = span{(1, 0, 1), (0, 1, 1)} v = (2, 3, 4)
T : R2 → R2, T (x, y) = (x, 1)
T : R3 → R3, T (x, y, z) = (x + y, x − y, z)
T : R3 → R3, T (x, y, z) = (x + 1, y + 1, z + 1)
T : M2,2 → R, T (A) = |A| = det A;  0 0 1  T : M3,3 → M3,3, T (A) = 0 1 0 A;   1 0 0 T : M T 2,2 → M2,2, T (A) = A T : R3 → R3
T (1, 0, 0) = (2, 4, −1), T (0, 1, 0) = (1, 3, −2) T (0, 0, 1) = (0, −2, 2) T (0, 3, −1) T (2, −1, 0) T : R3 → R3
T (1, 1, 1) = (2, 0, −1), T (0, −1, 2) = (−3, 2, −1) T (1, 0, 1) = (1, 1, 0) T (2, 1, 0) T (2, −1, 1)  1 2  T : R2 → R3 A = , T (2, 4)  −2 4  −2 2 T −1(−1, 2, 2).
T : R3 → R3, T (x, y, z) = (x, 0, z)
T : R3 → R3, T (x, y, z) = (z, y, x) ker T imT T T (v) = Av  1 −1 2   4 1  A = ; 0 1 2 A =  0 0  ; 2 −3  1 2  A = 1 2  1 1 0 0   − −  ; A = . 1 1 0 0 1 1 ker T T T
T : R2 → R2, T (x, y) = (x + 2y, x − 2y)
T : R3 → R3, T (x, y, z) = (2x − 3y, x − y, z)
T : R3 → R3, T (x, y, z) = (0, 0, 0)
T : R4 → R2, T (x1, x2, x3, x4) = (x1 + x2, x3 + x4) T = T2 ◦ T1
T1 : R2 → R2, T1(x, y) = (x − 2y, 2x + 3y),
T2 : R2 → R2, T2(x, y) = (2x, x − y);
T1 : R2 → R3, T1(x, y) = (x − 2y, x + y, x − y),
T2 : R3 → R2, T2(x, y, z) = (x − 3y, 3x + z);
T1 : R3 → R2, T1(x, y, z) = (x − 3y, 3x + z),
T2 : R2 → R3, T2(x, y) = (x − 2y, x + y, x − y).
T : R3 → R3, T (x, y, z) = (x, x + y, x + y + z)
T : R3 → R3, T (x, y, z) = (x + y, y + z, x + z)
T : R4 → R4, T (x1, x2, x3, x4) = (x1 − 2x2, x2, x3 + x4, x3)
T : R4 → R4, T (x1, x2, x3, x4) = (x4, x3, x2, x1) A T B, B′ [v]B [T (v)]B′ T (v) B′ [T (v)]B′ = A[v]B
T : R2 → R2, T (x, y) = (2x − 12y, x − 5y), v = (10, 5), B = B′ = {(4, 1), (3, 1)};
T : R2 → R3, T (x, y) = (x + y, x, y), v = (5, 4),
B = {(1, −1), (0, 1)}, B′ = {(1, 1, 0), (0, 1, 1), (1, 0, 1)};
T : R3 → R2, T (x, y, z) = (x − y, y − z), v = (1, 2, −3),
B = {(1, 1, 1), (1, 1, 0), (0, 1, 1)}, B′ = {(1, 2), (1, 1)};
T : R3 → R3, T (x, y, z) = (x + y + z, −x + 2z, 2y − z), v = (4, −5, 10),
B = {(2, 0, 1), (0, 2, 1), (1, 2, 1)}, B′ = {(1, 1, 1), (1, 1, 0), (0, 1, 1)}. T : R3 → R3
T (x, y, z) = (x + y + z, −x + 2y + 3z, 2x − y + z). T R3 T
{(1, 2, −1), (1, 0, 0), (0, 1, 0)} R3 A′ T B′ A′ A T B P B′ B P −1 A′ = P −1AP
T : R2 → R2, T (x, y) = (2x − y, −x + y), B′ = {(1, −2), (0, 3)};
T : R3 → R3, T (x, y, z) = (x, y, z),
B′ = {(1, 1, 0), (1, 0, 1), (0, 1, 1)};
T : R3 → R3, T (x, y, z) = (x, x + 2y, x + y + 3z),
B′ = {(1, −1, 0), (0, 0, 1), (0, 1, −1)}.  3 2  B = {(1, 3), (−2, −2)} B′ = {(−12, 0), (−4, 4)} R2 A = 0 4 T : R2 → R2 B P B′ B   A P [v] −1 B [T (v)]B [v] . B′ = 2
[v]B = P [v]B′, [T (v)]B = A[v]B = AP [v]B′ A′ T : R2 → R2 B′ P −1 [T (v)]B′ [T (v)]B′ = A′[v]B′ [T (v)]B′ = P −1[T (v)]B 2
B = {(1, 1), (−2, 3)}, B′ = {(1, −1), (0, 1)}  3 2   1  A = , [v] . 0 4 B′ = −3 2
B = {(1, 2), (−1, −1)}, B′ = {(−4, 1), (0, 2)}  2 1   −1  A = , [v] . 0 −1 B′ = 4 2
B = {(1, −1), (−2, 1)}, B′ = {(−1, 1), (1, 2)}  2 1   1  A = , [v] . 0 −1 B′ = −4 2
B = {(1, 1, 0), (1, 0, 1), (0, 1, 1)}, B′ = {(1, 0, 0), (0, 1, 0), (0, 0, 1)}  3 −1 −1   1  2 2 A = , .  − 1 2 1 [v] 0 2 2  B′ =   1 1 5 −1 2 2 2
B = {(1, 0, 0), (0, 1, 0), (0, 0, 1)}, B′ = {(1, 1, −1), (1, −1, 1), (−1, 1, 1)}  3 −1 −1   2  2 2 A = , .  − 1 2 1 [v] 1 2 2  B′ =   1 1 5 1 2 2 λi xi  1 0  λ A = , 1 = 1, x1 = (1, 0), 0 −1 λ2 = −1, x2 = (0, 1);  2 3 1  λ1 = 2, x1 = (1, 0, 0), A = 0 , λ  −1 2  2 = −1, x2 = (1, −1, 0), 0 0 3 λ3 = 3, x3 = (5, 1, 2). xi A  7 2  A = , x , x , x , x 2 4 1 = (1, 2) 2 = (2, 1) 3 = (1, −2) 4 = (−1, 0);  −1 −1 1  A = −2 0 −2 , x , −4, 6), x , 0, 6), x , 2, 0), x , , 1).   1 = (2 2 = (2 3 = (2 4 = (−1 0 3 −3 1  6 −3     1 2 −2  A = −5 0 0 −2 1 A = 3 7 0   A =  −2 5 −2  4 −2 3 −6 6 −3  7 2      A = 2 −2 3 3 2 −3 2 4 A =  0 3 −2  A =  −3 −4 9  0 −1 2 −1 −2 5  2 0 1   3 2 1  A =  0 3 4  A =  0 0 2  0 0 1 0 2 0 A P −1AP  1 3   3 1  A = , P = −1 5 1 1  −1 1 0   0 1 −3  A = 0 3 0 , P = 0 4 0     4 −2 5 1 2 2  0 0   1  A = −2 1 2 0 A = 0 1 4   0 0 2    2 1 −1 1 1  A = 2 A = 0 −2 −1  −1 2  0 0 −1 A P A P −1AP P −1AP  6 −3     1 2 −2  A = −5 0 0 −2 1 A = 3 7 0 2 5 2   A =  − −  4 −2 3 −6 6 −3  7 2      A = 2 −2 3 3 2 −3 2 4 A = 0 3 −2   A =  −3 −4 9  0 −1 2 −1 −2 5  2 0 1   3 2 1  A = 0 3 4 A = 0 0 2     0 0 1 0 2 0 B T B
T : R2 → R2, T (x, y) = (x + y, x + y)
T : R3 → R3, T (x, y, z) = (−2x + 2y − 3z, 2x + y − 6z, −x − 2y)  1 0 0   3 0 0  A = 0 2 0 0 2 0 .  , B =    A B 0 0 3 0 0 1 P B = P −1AP.  −1 a −1  A =  −3 5 −1  a −3 3 1 a 2 A a = 3 P P −1AP  6 −2   1 3  A = ; A = ; −2 1 2 4  2 −2 1   −5 3 4  A = 3 7 −2 .  −2 3 4  ; A =   0 4 1 4 −2 3  1 3   0 2 2   2 −1 −1  A = 3 1 A = 2 0 2 A =    −1 2 −1  2 2 0 −1 −1 2  0 2   0 4 4    A = 3 0 0 2 0 A = 4 2 0   A = 0 1 0   4 0 −2 0 0 1  2 1 1   0 1 1  A = 1 2 1 A = 1 0 1     1 1 2 1 1 0 √ √  2 2   −4 0 3  A = A = 0 1 0  2 2 √ √     2 2  3 0 4 − 2 2    / 2/3 −2/3  −4/5 0 3 5 A = A = 0 1 0 2/3 1/3   3/5 0 4/5 P P T AP P T AP  1 1   0 3 0   1 1 0 0  A = 1 1 A =  3 0 4  1 1 0 0 A =   0 4 0  0 0 1 1    0 0 1 1    1 4 2  −1 2 A = A = 2 4  −1 1 2  2 2 2  1 a 0  A =  a 1 0  a 0 0 −3 A a A 3 a = 2 P P tAP x2 + y2 9x2 + 10y2 − 5xy
x2 + 2y2 + 3z2 − 2xy − 4xz − 5yz
x2 + 2x2 + 3z2 + 4t2 + xy − 2xz + 6xt − 4yz − 8yt A P P T AP A 2x2 − 2y2 − 3xy 5x2 + 5y2 − 2xy 16x2 + 9y2 + 24xy xy, xz, yz x2 + y2 + 4xy − 9 = 0 2x2 + 5y2 − 4xy − 36 = 0 xy + x − 2y + 3 = 0 √ 5x2 + 5y2 − 2xy + 10 2x = 0
3x2 + 3y2 + 8z2 − 2xy − 16 = 0
2x2 + 2y2 + 2z2 + 2xy + 2xz + 2yz − 1 = 0
x2 + y2 + z2 + 2xy − z − 8 = 0

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