I know how to approach the beginning of this question. Given that the linear mapping:
$L: \Bbb R^2 \to \Bbb R^3 $
with null space:
$\operatorname{null}(L) = \operatorname{span}\left\{\begin{bmatrix}1\\1\end{bmatrix}\right\}$
and range:
$\operatorname{range}(L) = \operatorname{span}\left\{\begin{bmatrix}1\\2\\3\end{bmatrix}\right\}$
Find the appropriate matrix.
I know that the range must be included in the final matrix, so that will give me:
$\begin{bmatrix}1 & ?\\2 & ?\\3 & ?\end{bmatrix}$
But that is as far as I have gotten. Can someone explain what to do from here forward?
Best Answer
The nullspace of a matrix $T$ is the set of all vectors $x$ such that $Tx=0$. Matrix multiplication of a matrix and a column vector can be represented by the dot product of the rows of the matrix and the vector like so
$$Tx = \begin{bmatrix} t_{11} & t_{12} & \cdots & t_{1m} \\ t_{21} & t_{22} & \cdots & t_{2m} \\ \vdots & \vdots & \cdots & \vdots \\ t_{n1} & t_{n2} & \cdots & t_{nm} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_m\end{bmatrix} = \begin{bmatrix} \text{row}_1 \\ \text{row}_2 \\ \vdots \\ \text{row}_n \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_m\end{bmatrix} = \begin{bmatrix} \text{row}_1 \cdot x \\ \text{row}_2 \cdot x \\ \vdots \\ \text{row}_n \cdot x \end{bmatrix}$$
Therefore all you need to do is solve $(1,?)\cdot (1,1)=0$, $(2,?)\cdot (1,1)=0$, and $(3,?)\cdot (1,1) = 0$.