it is a maths question algebra

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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