Find a linear transformation $T: \mathbb{R}^3 \to \mathbb{R}^3$ whose image is spanned by $(1,2,3)$ and $(4,5,6)$

Question

Here's the problem:

Find a linear map $T: \mathbb{R}^3 \to \mathbb{R}^3$ such that the image is spanned by $(1,2,3)$ and $(4,5,6)$.

Here's my work:

If we think of a matrix for $T$, say $[T]$, then we want the column space to be spanned by $(1,2,3)$ and $(4,5,6)$, since $\mathrm{Im}(T) =$ columnspace$([T])$. So any matrix with columns in $\{(1,2,3), (4,5,6)\}$ should work, e.g.

$[T] = \begin{pmatrix} 1 & 4 & 1\\ 2 & 5 & 2 \\ 3 & 6 & 3 \end{pmatrix}$.

Correct? Thanks!

Answer 1

Yes. You can completely categorize a linear transformation by its action on a basis set of the domain space. And since $T(1,0,0)=T(0,0,1)=(1,2,3)$ and $T(0,1,0)=(4,5,6)$ by your matrix definition, the range of $T$ will be the span of the two vectors.