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featheredteap
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Homework Statement
Let T be the linear transformation T: M2x2-->M2x2 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 of T and the nullspace of T.
Homework 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)
The Attempt at a Solution
I multiplied the elements of the standard basis by T to 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 matrix T relative to the standard basis S:
[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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