(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let T be the linear transformationT:M_{2x2}-->M_{2x2}given by

T([a,b;c,d]) = [a,b;c,d][0,0;1,1] = [b,b;d,d]

Find bases (consisting of 2x2 matrices) for the image ofTand the nullspace ofT.

2. Relevant equations

Standard basis of a 2x2 matrix: {[1,0;0,0],[0,1;0,0],[0,0;1,0],[0,0;0,1]}

rank(T) + nullity (T) = n (number of columns of T)

3. The attempt at a solution

I multiplied the elements of the standard basis byTto find the image points of the transformation. I then put those image points in the form [a,b;c,d] in a matrix, which equalled the matrixTrelative to the standard basisS:

[T]_{S}=[0,0,0,0;0,1,0,0;0,0,0,1;0,0,0,0]

To find a basis for the image, I took the columns with leading entries, but I'm not completely sure it's correct:

basis: {[0,1;0,0],[0,0;1,0]}

As for the basis for the nullspace, wouldn't it just be {0} because there is only one solution to the system of equations (i.e. they are linearly independent)? Or is it [0,0;0,0]? However, rank + nullity = 4 and (assuming I got the basis for the image right) rank = 2, so nullity should = 2. Does the nullity of a 2x2 matrix = 2?

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# Homework Help: Image and nullspace bases of a linear transformation

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