Find all Linear Transformations with specified Image and Kernel

Question

$T: R^3 -> R^3$

Find all the linear transformations such that:

1. The vectors

$$v = \left( \begin{array}{c} 1\\ -1\\ 0\\ \end{array} \right)$$

$$w = \left( \begin{array}{c} 0\\ 1\\ 1\\ \end{array} \right)$$

are a basis for the kernel of $T$ and

2. The image of $T$ is a line that has a direction vector

$$s = \left( \begin{array}{c} 0\\ 0\\ 1\\ \end{array} \right)$$

Answer 1

Hint: Extend $\{w, v\}$ to a basis of $\mathbb R^3$. You know every linear map is uniquely identified by where it sends a basis and you already know $w$ and $v$ should go to zero. Your third basis element must then get sent to something on the line with direction vector $s$.

Answer 2

Hint
You are given the equations $$Tx = (0,0,s)^T \qquad \forall x$$ Wich already necessitates $T_{ij} = 0 \forall i=1,2; j=1..3$. $$T = \pmatrix{0&0&0\\0&0&0\\&t^T&}$$ And for $t$ you are given the equations $$t^T v = t^T w = 0$$ This will give you a one-dimensional solution space with basis $(1,1,-1)^T$, so $$T = \lambda \pmatrix{0&0&0\\0&0&0\\1&1&-1}$$