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Let $$T\colon V\to W$$ be a linear transformation defined by the matrix A where $$A= \left[ \begin{matrix} -1 & 2 & 3 \\ 0 & 1 & 0 \\ 1 & -1 & -2 \\ 0 & 0 & 1 \\ \end{matrix} \right] $$

I have to find basis for $V = R^3$ and $W = R^4$ to make the matrix of T as simple as possible.

How would I go about doing this? I've never done something like this before and I'm not sure where to start.

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  • $\begingroup$ What do you mean "find basis given the matrix" ? Their bases should not depend on the transformation. $\endgroup$
    – User
    Mar 7, 2017 at 11:16
  • $\begingroup$ Accidentally left something out so I edited the question to make it clearer. $\endgroup$
    – Fate
    Mar 7, 2017 at 11:23
  • $\begingroup$ How about you take the columns of the matrix, and add one vector to complete that to a basis of $\Bbb{R}^4$? Then the matrix will look like the 3-by-3 identity matrix with a row of zeros underneath it. $\endgroup$
    – Nick
    Mar 7, 2017 at 16:20

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