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!


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

  • $\begingroup$ 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. $\endgroup$ 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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