maths9650
ssttuuddeennttlinier_algebra_proofs.pdf
MATH 2107 - Assignment 1 Due June 9 at the beginning of tutorial
1. (5 points) Let T : Mnn(R) → R defined by T (A) = rank(A). Is T a linear transformation? If it is, show it; otherwise, provide an example to disprove it.
2. (5 points) Let W be a subspace of a finite-dimensional vector space V . Prove that dimW = dimV if and only if W = V .
3. (10 points) Recall that W = { A ∈ M22(R); AT = A
} is a subspace of M22(R).
Define a linear transformation T : W → P2 by T ( a b c d
) = (a − b) + (c − d)x +
(d−a)x2. Find rank(T ) by using the rank theorem for T .
4. (10 points) Let B and C be bases for R2. If C = {(
1 2
) ,
( 2 3
)} and the change-
of-coordinates matrix from B to C is PC←B = (
1 −1 −1 2
) , find B.
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