Linear Algebra

248 Chapter 3. Maps Between Spaces
�
1
(c)
2
� �
3 1
,
6 2
�
3
−6
� 2.11 Find the canonical representative of the matrix-equivalence class of each matrix.
(a)
�
2
4
1
2
�
0
0
(b)
�
0
1
3
1
1
3
0
0
3
�
2
4
−1
2.12 Suppose that, with respect to
B = E2
� � � �
1 1
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.
(a) ˆ � � � �
0 1
B = 〈 , 〉,
1 1
ˆ � � � �
−1 2
D = 〈 , 〉
0 1
(b) ˆ � � � �
1 1
B = 〈 , 〉,
2 0
ˆ � � � �
1 2
D = 〈 , 〉
2 1
� 2.13 What size are P and Q?
� 2.14 Use Theorem 2.6 to show that a square matrix is nonsingular if and only if it
is equivalent to an identity matrix.
� 2.15 Show that, where A is a nonsingular square matrix, if P and Q are nonsingular
square matrices such that PAQ = I then QP = A −1 .
� 2.16 Why does Theorem 2.6 not show that every matrix is diagonalizable (see
Example 2.2)?
2.17 Must matrix equivalent matrices have matrix equivalent transposes?
2.18 What happens in Theorem 2.6 if k =0?
� 2.19 Show that matrix-equivalence is an equivalence relation.
� 2.20 Show that a zero matrix is alone in its matrix equivalence class. Are there
other matrices like that?
2.21 What are the matrix equivalence classes of matrices of transformations on R 1 ?
R 3 ?
2.22 How many matrix equivalence classes are there?
2.23 Are matrix equivalence classes closed under scalar multiplication? Addition?
2.24 Let t: R n → R n represented by T with respect to En, En.
(a) Find RepB,B(t) in this specific case.
� � � � � �
1 1
1 −1
T =
B = 〈 , 〉
3 −1
2 −1
(b) Describe Rep B,B(t) in the general case where B = 〈 � β1,... , � βn〉.
2.25 (a) Let V have bases B1 and B2 and suppose that W has the basis D. Where
h: V → W , find the formula that computes Rep B2,D(h) fromRep B1,D(h).
(b) Repeat the prior question with one basis for V and two bases for W .
2.26 (a) If two matrices are matrix-equivalent and invertible, must their inverses
be matrix-equivalent?
(b) If two matrices have matrix-equivalent inverses, must the two be matrixequivalent?