Page 276 - 35Linear Algebra

276 Orthonormal Bases and Complements

12. For U a subspace of W, use the subspace theorem to check that U ⊥ is
a subspace of W.

13. Let S n and A n define the space of n×n symmetric and anti-symmetric
n
matrices, respectively. These are subspaces of the vector space M of
n
n
all n × n matrices. What is dim M , dim S n , and dim A n ? Show that
n
n
M = S n + A n . Define an inner product on square matrices
n
M · N = trMN .
n
⊥
Is A = S n ? Is M = S n ⊕ A n ?
n n
14. The vector space V = span{sin(t), sin(2t), sin(3t), sin(3t)} has an inner
product:
Z 2π
f · g := f(t)g(t)dt .
0
Find the orthogonal compliment to U = span{sin(t) + sin(2t)} in V .
⊥
Express sin(t) − sin(2t) as the sum of vectors from U and U .

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