Page 223 - 35Linear Algebra

11.3 Review Problems 223

6. Let S n denote the vector space of all n × n symmetric matrices;

T
n
n
S n := {M : R → R | M = M }.
Let A n denote the vector space of all n × n anti-symmetric matrices;

T
n
n
A n = {M : R → R | M = −M }.
(a) Find a basis for S 3 .
(b) Find a basis for A 3 .
(c) Can you find a basis for S n ? For A n ?
Hint: Describe it in terms of combinations of the matrices F j i
which have a 1 in the i-th row and the j-th column and 0 every-
i
where else. Note that {F | 1 ≤ i ≤ r, 1 ≤ j ≤ k} is a basis for
j
r
M .
k
7. Give the matrix of the linear transformation L with respect to the input
0
and output bases B and B listed below:
0
(a) L : V → W where B = (v 1 , . . . , v n ) is a basis for V and B =
(L(v 1 ), . . . , L(v n )) is a basis for W.
0
(b) L : V → V where B = B = (v 1 , . . . , v n ) and L(v i ) = λ i v i for
all 1 ≤ i ≤ n.

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