Linear Algebra, 2020a

Section V. Change of Basis 273
Exercises
̌ 2.11 Decide
(
if these
) (
are matrix
)
equivalent.
1 3 0 2 2 1
(a)
,
2 3 0 0 5 −1
( ) ( )
0 3 4 0
(b) ,
1 1 0 5
( ) ( )
1 3 1 3
(c) ,
2 6 2 −6
2.12 Which of these are matrix equivalent to each other?
⎛ ⎞
1 2 3 ( ) ( ) ( )
(a) ⎝4 5 6⎠
1 3
−5 1 0 0 −1
(b)
(c)
(d)
−1 −3 −1 0 1 0 5
7 8 9
⎛ ⎞ ⎛
⎞
1 0 1
3 1 0
(e) ⎝2 0 2⎠
(f) ⎝ 9 3 0⎠
1 3 1 −3 −1 0
̌ 2.13 Find the canonical representative of the matrix equivalence class of each matrix.
⎛
⎞
( ) 0 1 0 2
2 1 0
(a)
(b) ⎝1 1 0 4 ⎠
4 2 0
3 3 3 −1
2.14 Suppose that, with respect to
( ( 1 1
B = E 2 D = 〈 , 〉
1)
−1)
the transformation t: R 2 → R 2 is represented by this matrix.
( ) 1 2
3 4
Use change
(
of basis
(
matrices
(
to represent
) (
t with respect to each pair.
0 1 −1 2
(a) ˆB = 〈 , 〉, ˆD = 〈 , 〉
1)
1)
0 1)
( ( ( ( 1 1 1 2
(b) ˆB = 〈 , 〉, ˆD = 〈 , 〉
2)
0)
2)
1)
2.15 What sizes are P and Q in the equation Ĥ = PHQ?
̌ 2.16 Consider the spaces V = P 2 and W = M 2×2 , with these bases.
( ) ( ) ( ) ( )
0 0 0 0 0 1 1 1
B = 〈1, 1 + x, 1 + x 2 〉 D = 〈 , , , 〉
0 1 1 1 1 1 1 1
( ) ( ) ( ) ( )
−1 0 0 −1 0 0 0 0
ˆB = 〈1, x, x 2 〉 ˆD = 〈 , , , 〉
0 0 0 0 1 0 0 1
We will find P and Q to convert the representation of a map with respect to B, D
to one with respect to ˆB, ˆD
(a) Draw the appropriate arrow diagram.
(b) Compute P and Q.
̌ 2.17 Find the change of basis matrices Q and P that will convert the representation
of a t: R 2 → R 2 with respect to B, D to one with respect to ˆB, ˆD.
( ( ( ( )
( ( 1 1 0 −1
1 0
B = 〈 , 〉 D = 〈 , 〉 ˆB = E 2
ˆD = 〈 , 〉
0)
1)
1)
0
−1)
1)