Linear Algebra, Theory And Applications, 2012a

488 ANSWERS TO SELECTED EXERCISES
10 (Ax, y) = ∑ i (Ax) i y i = ∑ ∑
i k A ikx k y i
(
x,A T y ) = ∑ k x ∑ ( )
k i A
T
ki y i = ∑ ∑
k i x kA ik y i ,
the same as above. Hence the two are equal.
( )
11 (AB) T x, y ≡
(x, (AB) y) =
(
A T x,By ) = ( B T A T x, y ) (
. Since this holds for every
x, y, you have for all y, (AB) T x − B T A T x, y
)
.
Let y =(AB) T x − B T A T x. Then since x is arbitrary,
the result follows.
13 Give an example of matrices, A, B, C such that B ≠
C, A ≠0, and yet AB = AC.
( )( ) ( )
1 1 1 −1 0 0
=
1 1 −1 1 0 0
( )( ) ( )
1 1 −1 1 0 0
=
1 1 1 −1 0 0
15 It appears that there are 8 ways to do this.
17 ABB −1 A −1 = AIA −1 = I
B −1 A −1 AB = B −1 IB = I
Then by the definition of the inverse and its uniqueness,
it follows that (AB) −1 exists and (AB) −1 =
B −1 A −1 .
19 Multiply both sides on the left by A −1 .
( )( ) ( )
1 1 1 −1 0 0
21
=
1 1 −1 1 0 0
23 Almost anything works.
( )( ) ( )
1 2 1 2 5 2
=
3 4 2 0 11 6
( )( ) ( )
1 2 1 2 7 10
=
2 0 3 4 2 4
( )
−z −w
25
,z,w arbitrary.
z w
27
⎛
⎝ 1 2 2 1 3
4
⎞
⎠
1 0 2
−1
29 Row echelon form:
⎛
⎝ 1 0 ⎞
5
3
2
0 1 ⎠
3
.Ahas no in-
0 0 0
verse.
⎛
= ⎝
−2 4 −5
0 1 −2
1 −2 3
⎞
⎠
G.6 Exercises
2.7
1 Show the map T : R n → R m defined by T (x) =Ax
where A is an m×n matrix and x is an m×1 column
vector is a linear transformation.
This follows from matrix multiplication rules.
3 Find the matrix for the linear transformation which
rotates every vector in R 2 through an angle of π/4.
( ) (
cos (π/4) − sin (π/4)
1
√ )
= 2 2 −
1
√ 2√
√ 2
sin (π/4) cos (π/4)
1
2 2
1
2 2
5 Find the matrix for the linear transformation which
rotates every vector in R 2 throughanangleof2π/3.
( ) ( √ )
2 cos (π/3) −2sin(π/3)
= √ 1 − 3
2sin(π/3) 2 cos (π/3) 3 1
7 Find the matrix for the linear transformation which
rotates every vector in R 2 through an angle of 2π/3
and then reflects across the x axis.
( )( )
1 0 cos (2π/3) − sin (2π/3)
0 −1 sin (2π/3) cos (2π/3)
( )
−
1
= 2
− 2√ 1 3
− 2√ 1 3
1
2
9 Find the matrix for the linear transformation which
rotates every vector in R 2 through an angle of π/4
and then reflects across the x axis.
( )( )
1 0 cos (π/4) − sin (π/4)
0 −1 sin (π/4) cos (π/4)
( 1
√ )
= 2 2 −
1
2√
√ 2
− 2√ 1 2 −
1 2
2
11 Find the matrix for the linear transformation which
reflects every vector in R 2 across the x axis and then
rotates every vector through an angle of π/4.
( )( )
cos (π/4) − sin (π/4) 1 0
sin (π/4) cos (π/4) 0 −1
( 1
√ )
= 2 2
1
√ 2√
√ 2
1
2 2 −
1
2 2
13 Find the matrix for the linear transformation which
reflects every vector in R 2 across the x axis and then
rotates every vector through an angle of π/6.
( )( )
cos (π/6) − sin (π/6) 1 0
sin (π/6) cos (π/6) 0 −1
√
3
1
=
( 1
2
2
1
2
− 1 2
√
3
)