n
(n 1)
A
=
1 2 1
1 1 0
2 0 1
A
100
2x
1
+ 3 + 2x
2
+ x
3
x
4
= 3
4x
1
+ 6 + 3 + 4x
2
x
3
x
4
= 5
6x
1
+ 9 + 5 + 6x
2
x
3
x
4
= 7
8x
1
+ 12 + 7 + 8x
2
x
3
x
4
= 9
E
1
=
1 0
0 0
E
2
=
0 1
0 0
E
3
=
0 0
1 0
E
4
=
0 0
0 1
R
Mat(2; )R
A
=
a b
C d
ϕ : Mat Mat(2; R) (2; )R
ϕ(X) = X.A
ϕ
{ }E
1
, E , E , E
2 3 4
A
a 6= 0
Im ϕ
Ker ϕ
A
+1 1
det A
det A 4

Preview text:

n (n ≥ 1)  1 −2 1  A = −1 1 0   −2 0 1 A100  2x1 + 3x2 + x3 + 2x4 = 3    4x1 + 6x2 + 3x3 + 4x4 = 5 6x1 + 9x2 + 5x3 + 6x4 = 7    8x1 + 12x2 + 7x3 + 8x4 = 9  1 0   0 −1   0 0  E1 = E E 0 0 2 = 0 0 3 = 1 0  0 0  E4 = 0 1 R M at(2; R)  a b  A = C d ϕ : M at(2; R) → Mat(2; R) ϕ(X) = X.A ϕ {E1, E2, E3, E4} A a 6= 0 Im ϕ Ker ϕ A +1 −1 det A det A ≤ 4