Getting the image: involves LT composition and change of basis matrix.

Hi!

i have a doubt here:

Excercise:

Given:

Both are basis from the V linear space.

and

-->Find

So, my plan was:

Get the change of basis matrix, this is easy because of the way the basis are expresed

Now i know that

Now i just do

And those columns are the image of but expresed in , so, image well expresed is

butīs LD, so

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So, the problem is that in the book i found a proposition that says that

given linear functions and

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So, that, what i have done is:

and i know now that the image of the LT are the colums of the matrix, but those are expresed in so, the image is going to be formed by:

butīs LD, so

Unfortunately, both results are different!!

which is the correct one? where is the mistake?