Finding the transformation matrix relative to bases..

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Posted by Bradley Vrecko on November 05, 2002 at 23:12:06:

I am hoping some one can shed some light on this question for me:

Find the B-matrix (B is actually a beta symbol) of the transformation mapping x to Ax.
Matrix A is 2X2 = [3 4, -1 -1] and the basis vectors are 2X1, b1 = [2, -1] and b2 = [1, 2]

OK, so my understanding is that the Beta matrix signifies that the transformation from a vector space V to W is all in the same basis. So mapping x to T(x) by matrix A is very similar to mapping [x]B to [T(x)]B by the Beta matrix.

With the information listed, how do I go about this? Here is what I tried. First, I tried to diagonalize A, but it only has one distinct eigenvector so it is not diagonalizable.

Then I let the vector x = [1, 1] and found T(x) by multiplying x by Ax. I then used the vector x and found the its representation in the Beta coordinates by since I know that x can be written as a linear combination of basis vectors, with the weights being [x]B.

So now I have [x]B, and doing the same calculations I changed T(x) into [T(x)]B. Since the Beta matrix maps [x]B to [T(x)]B, and I have both those values, I should be able to find the Beta matrix of transformation. But I do not get the correct answer.

Please post up if you have any ideas, I am not opposed to trying different things--even if you are not sure it will work. Obviously I would prefer someone to give me the correct method in solving these types of problems, but after struggling with this one I am open to anything. :)

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