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Môn: Đại số tuyến tính( MATH 231A)

Trường: Đại học Sư Phạm Hà Nội

Thông tin:

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Tác giả:

Profile Kim Oanh
Linear Algebra
Exercises and Problems
Linear system
1. Find all to the following in form:
(a) 2x + 3y = 1;
(b) x − 2y + 5z = 1;
(c)
x − y = 3
y
+ 2z = 4
.
2.
Find the solution of each of the following using
representation.
(a)
x + 2y = 1
3
x + 4y = −1
;
(b)
3x + 4y = 1
4
x + 5y = −3
;
(c)
x + y + 2z = −1
2x + y + 3z = 0
−2y + z = 2
;
(d)
2x + y + z = −1
x + 2y + z = 0
3x − 2z = 5
.
3. Find all solution (if any) to each of the following .
(a)
− −2x + 3 + 3y z = 9
3x − 4y + z = 5
− −5x + 7 + 2y z = 14
;
(b)
x + 2y − z = 2
2x + 5y − 3z = 1
x + 4y − 3z = 3
;
(c)
x x
1
− x
2
+ x
3
−
4
= 0
−x x x x
1
+
2
+
3
+
4
= 0
x x x
1
+
2
− x
3
+
4
= 0
x x x
1
+
2
+
3
+ x
4
= 0
;
(d)
x
1
+ x x x
2
+ 2
3
−
4
= 4
3x x
2
−
3
+ 4x
4
= 2
x
1
+ 2x x
2
− 3x
3
+ 5
4
= 0
x x
1
+ x
2
− 5x
3
+ 6
4
= 0
.
4. Find the circle passing through the following points:
(a) (−2, 1) (5 1);, , 0) and (4,
(b) (1, 1) (5 3)., , −3) and (−3, −
1
A B
C
D
5. The Simpson’s formula is an equality of the form
Z
1
−1
f(t)dt = af bf cf(−1) + (0) + (1)
for any polynomial
f
of degree at most 3 and
a, b, c
are constants. Find these constants.
6.
Three Nissans, two Fords, and four Chevrolets can be rented for $106 per day. At the
same rates two Nissans, four Fords, and three Chevrolets cost $107 per day, whereas
four Nissans, three Fords, and two Chevrolets cost $102 per day. Find the rental rates
for all three kinds of cars.
7.
A boy finds $1
.
05 in dimes, nickels, and pennies. If there are 17 coins in all, how many
coins of each type can he have?
8.
The scores of three players in a tournament have been lost. The only information
available is the total of the scores for players 1 and 2, the total for players 2 and 3,
and the total for players 3 and 1.
(a) Show that the individual scores can be rediscovered.
(b)
Is this possible with four players (knowing the totals for players 1 and 2, 2 and 3,
3 and 4, and 4 and 1)?
9.
A network of one-way streets is shown in the diagram. The rate of flow of cars into
intersection A is 500 cars per hour, and 400 and 100 cars per hour emerge from B and
C, respectively. Find the possible flows along each street.
(Hint: The Junction rule is that at each of the junctions in the network, the total flow
into that junction must equal the total flow out.)
10.
Show that every elementary row operation can be reversed by another one of the same
type (called its inverse).
Hence prove that elementary operations keep the set of solutions of linear systems be
the same.
Vector space/subspace
11. Let V and W be real vector spaces.
(a) Show that V × W is a real vector space with the following operations
(~v
1
, ~w
1
) + (~v
2
, ~w
2
) = ( ~v ~v
1
+
2
, ~w
1
+ ~w ,
2
)
λ(~v
1
, ~w
1
) = (λ ~v
1
, λ ~w ,
1
)
with ~v
1
, ~v
2
∈ V , ~w
1
, ~w
2
∈ W and λ ∈ R.
2
(b) Show that V × V is a complex vector space with the following operations
(~v
1
, ~v
2
) + (~v
1
0
, ~v
2
0
) = ( ~v ~v
1
+
1
0
, ~v
2
+ ~v
2
0
),
(a + bi ~v)(
1
, ~v
2
) = (a ~v b ~v
1
−
2
, a ~v
2
+ b~v
1
)
with ~v
1
, ~v , ~v
2 1
0
, ~v
2
0
∈ ∈V and a, b R.
12.
Let
P
3
denote the set of all polynomials with real coefficients of degree at most 3. Show
that
P
3
along with the addition of polynomials and the multiplication of polynomials
by real numbers is a real vector space.
13. In Q we consider the following two operations:
⊕ : r ⊕ s = r + s,
: λ r =
λr if ;λ ∈ Q
0 if λ /∈ Q.
Is (Q, ⊕, ) a real vector space? Justify your answer.
14.
Show that the last axiom in the definition of vector spaces can be replaced by the
following condition:
λ~x = 0 ⇔ λ = 0 or ~x .= 0
3
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Linear Algebra Exercises and Problems Linear system 1. Find all to the following in form: (a) 2x + 3y = 1; (b) x − 2y + 5z = 1;  x − y = 3 (c) . y + 2z = 4
2. Find the solution of each of the following using representation.  x + 2y = 1 (a) ; 3x + 4y = −1  3x + 4y = 1 (b) ; 4x + 5y = −3  x + y + 2z = −1  (c) 2x + y + 3z = 0 ;  −2y + z = 2  2x + y + z = −1  (d) x + 2y + z = 0 .  3x − 2z = 5
3. Find all solution (if any) to each of the following .  −2x + 3y + 3z = −9  (a) 3x − 4y + z = 5 ;  −5x + 7y + 2z = −14  x + 2y − z = 2  (b) 2x + 5y − 3z = 1 ;  x + 4y − 3z = 3  x1 − x2 + x3 − x4 = 0    −x1 + x2 + x3 + x (c) 4 = 0 ; x1 + x2 − x3 + x4 = 0    x1 + x2 + x3 + x4 = 0  x1 + x2 + 2x3 − x4 = 4    3x2 − x (d) 3 + 4x4 = 2 . x1 + 2x2 − 3x3 + 5x4 = 0    x1 + x2 − 5x3 + 6x4 = 0
4. Find the circle passing through the following points:
(a) (−2, 1), (5, 0) and (4, 1);
(b) (1, 1), (5, −3) and (−3, −3). 1 A B D C
5. The Simpson’s formula is an equality of the form Z 1
f (t)dt = af (−1) + bf (0) + cf (1) −1
for any polynomial f of degree at most 3 and a, b, c are constants. Find these constants.
6. Three Nissans, two Fords, and four Chevrolets can be rented for $106 per day. At the
same rates two Nissans, four Fords, and three Chevrolets cost $107 per day, whereas
four Nissans, three Fords, and two Chevrolets cost $102 per day. Find the rental rates for all three kinds of cars.
7. A boy finds $1.05 in dimes, nickels, and pennies. If there are 17 coins in all, how many
coins of each type can he have?
8. The scores of three players in a tournament have been lost. The only information
available is the total of the scores for players 1 and 2, the total for players 2 and 3,
and the total for players 3 and 1.
(a) Show that the individual scores can be rediscovered.
(b) Is this possible with four players (knowing the totals for players 1 and 2, 2 and 3, 3 and 4, and 4 and 1)?
9. A network of one-way streets is shown in the diagram. The rate of flow of cars into
intersection A is 500 cars per hour, and 400 and 100 cars per hour emerge from B and
C, respectively. Find the possible flows along each street.
(Hint: The Junction rule is that at each of the junctions in the network, the total flow
into that junction must equal the total flow out.)
10. Show that every elementary row operation can be reversed by another one of the same type (called its inverse).
Hence prove that elementary operations keep the set of solutions of linear systems be the same. Vector space/subspace
11. Let V and W be real vector spaces.
(a) Show that V × W is a real vector space with the following operations ( ~ v1, ~ w1) + ( ~ v2, ~ w2) = ( ~ v1 + ~ v2, ~ w1 + ~ w2), λ( ~ v1, ~ w1) = (λ ~ v1, λ ~ w1), with ~ v1, ~ v2 ∈ V , ~ w1, ~ w2 ∈ W and λ ∈ R. 2
(b) Show that V × V is a complex vector space with the following operations ( ~ v 0 0 0 0 1, ~ v2) + ( ~ v1 , ~ v2 ) = ( ~ v1 + ~ v1 , ~ v2 + ~ v2 ), (a + bi)( ~ v1, ~ v2) = (a ~ v1 − b ~ v2, a ~ v2 + b ~ v1) with ~ v 0, ~ v 0 ∈ V and a, b ∈ R. 1, ~ v2, ~ v1 2
12. Let P3 denote the set of all polynomials with real coefficients of degree at most 3. Show
that P along with the addition of polynomials and the multiplication of polynomials 3
by real numbers is a real vector space.
13. In Q we consider the following two operations: ⊕ : r ⊕ s = r + s,  λr if λ ∈ Q;  : λ  r = 0 if λ / ∈ Q.
Is (Q, ⊕, ) a real vector space? Justify your answer.
14. Show that the last axiom in the definition of vector spaces can be replaced by the following condition: λ~x = 0 ⇔ λ = 0 or ~x = 0. 3