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Unformatted text preview: Math 235 Tutorial 3 Problems a1 a2
bb
,12
= a1 b1 + 2a2 b2 + 3a3 b3 is an inner product on the
0 a3
0 b3
vector space of 2 × 2 upper triangular matrices. 1: Prove that 0 x1
 x1 + x2 = x3
x2 x3
to T = {p(x) ∈ P2  p(1) = 0} to establish that the spaces are isomorphic. Prove that
your map is an isomorphism. 2: Deﬁne an explicit isomorphism from the vector space S = 3: Prove that if L : V → W and M : W → U are both onto linear mappings then
M ◦ L : V → U is onto.
4: Let V and W be ndimensional vector spaces with bases B and C respectively. Let
L : V → W be a linear mapping. Prove that L is an isomorphism if and only if the
matrix of L with respect to B and C has rank n. (Prove this without using the fact
that the rank of L equals the rank of any matrix of L.) 1 ...
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 Fall '08
 CELMIN
 Matrices, Vector Space

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