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Suppose $T:R^2 \to R^3$ is a linear transformation such that: $T([-1,0]) = (-1,1,0)$ and $T([2,-1]) = (3,-3,k)$.

Find the standard matrix $A$ for $T$. I have found this, it is

$$A= \begin{bmatrix} 1 & -1 \\ -1 & 1 \\ 0 & -k \end{bmatrix} $$ T is one-to-one only if k does not equal? (with proof). I have found this, $k$ cannot equal $0$, otherwise there exists a free variable.

When $k=0$, find all solutions to the equation $T(x)=0$. I end up with just the matrix:

$$\begin{bmatrix}1 &-1 \\0 & 0\\0 & 0 \end{bmatrix}$$ How do I write all solutions? Can I just write the column vector $(x2, x2)$?

When $k=1$, find all solutions to the equation $T(x)=0$. I end up with the matrix:

$$\begin{bmatrix}1& 0\\0 & 1\\0& 0 \end{bmatrix}$$ How do I write this as a general solution? Is it just the column vector $(0,0)$?

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    You may first want to write the standard basis vectors $e_{1} = \begin{bmatrix}1\\0\end{bmatrix}$ and $e_{2} = \begin{bmatrix}0\\1\end{bmatrix}$ as linear combinations of $\begin{bmatrix}-1\\0\end{bmatrix}$ and $\begin{bmatrix}2\\-1\end{bmatrix}$ . Then compute $T(e_{1})$ and $T(e_{2})$, and those will be the two columns of your matrix. – 2017-02-18
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    Very good so far! – 2017-02-18
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    @AndreasCaranti Thank you, can you help me with writing the general solutions for the last two? I think I have it , but I'm not sure. – 2017-02-18

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