Matrix column addition

Question

Suppose you have a matrix in the form of: $$\left[\begin{array}{c} a\\ b\end{array}\right]$$

How can this be represented be a two by two matrix?

Answer 1

For exmaple you can do the following: Note that $$ W:=\{\begin{pmatrix}a & 0\\ b & 0 \end{pmatrix}\mid a,b\in\mathbb{R}\} $$

is closed under addition and multiplication by scalar, since the zero vector is also in $W$, it follows that $W$ is a sub vector space of $M_{2}(\mathbb{R})$.

Clearly $$ \dim(W)=2 $$

and there is a natural way of defining an isomorphism $$ T:\,\mathbb{R}^{2}\to W $$

and you can represent $v$ as $T(v)$ which is a $2\times2$ matrix.

This can be done with any sub vector space of dimension $2$ of $M_{2}(\mathbb{R})$ in a similar manner