Hello, the problem I am having trouble with is below, Let D3 < M3x3 denote the s

Question

Hello, the problem I am having trouble with is below,

Let D3 < M3x3 denote the space of diagonal 3 x 3 matrices, and let T : P3 --> D3 be a linear transformation given by p(t) -->

P(0) 0 0
0 P(1) 0
0 0 P(2)

(a 3x3 matrix)

Determine the matrix A of the transformation T relative to the bases B = {1, t, t^2, t^3} for P3
and C=

{
1 0 0
0 0 0
0 0 0

0 0 0
0 1 0
0 0 0

0 0 0
0 0 0
0 0 1

}

for D3.
(C is set of three 3x3 vectors)

Determine bases for Nul(A) and Col(A), and Determine a basis for the kernel of T. what is range of T?

[Hint: what is the dimension of range(T)? what about dim(D3)?]

Thanks!

Explanation / Answer

Well notice that 1 is sent to the diagonal matrix with entries 1,1,1; t is sent to the diagonal matrix with entries 0,1,2 ; t^2 is sent to the diagonal matrix with entries 0,1^2, 2^2 this is 0,1,4 and finally t^3 is sent to the diagonal matrix with entries 0,1,8.

So the matrix A of the transformation T relative to the above basis is:

( 1,0,0,0 ; (new row) 1,1,1,1 ; (new row) 1,2,4,8).

Time is running out but for example to get the kernel of T, look at what p(t) would give you the zero matrix. This are the p(t) such that p(0)=p(1)=p(2)=0. So those are the polynomials gnerated by (x)(x-1)(x-2). The range of T is all 3 by 3 diagonal matrices.

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