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Let $B=[(1,0,0),(1,2,0),(1,2,3)]$ be the basis for $\mathbb{R}^3$.

Let $T:\mathbb{R}^3\rightarrow\mathbb{R}^3$ be a linear transformation, such that its representing matrix with respect to basis $B$ is the following:

$$ A= \begin{bmatrix} 1 & 2 & 1\\ 0 & -1 & 3\\ 4 & -2& 5 \end{bmatrix} $$

a. find the representing matrix for $T$ with respect to the standard basis.

b. is $T$ an isomorphism?

This is homework help and I would appriciate any ideas. I was thinking of finding the trasformations using the coordinates vector, but im not sure if it is the right idea since it looks like a lot of work.

Thanks in advance!

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Hint for a.

1)write down the images of basis vectors

2)write standard basis vectors in terms of given basis vectors

3)find the images of standard basis vectors

Hint for b.

$T$ is invertible iff its matrix wrt standard ordered basis is invertible

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  • $\begingroup$ In order to get the image of $(0,1,0)$ I will need to use the given matrix on the coordinates vector - is that correct or am I missing something? I guess im asking if I could bypass that. $\endgroup$ – Boris Ablamunits May 20 '14 at 10:06
  • $\begingroup$ you are correct.use given matrix on the coordinate vectors $\endgroup$ – usermath May 20 '14 at 10:14

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