How to determine if this isomorphism is a linear transformation?

Question

I am presented with the following isomorphism:

$$\begin{bmatrix}x & y\end{bmatrix}\longrightarrow\begin{bmatrix}x\\ y \end{bmatrix}$$

Is this a linear transformation and why?

Thank you for your help.

Answer 1

The mapping $T$ shows that $T(x,y)^{r} = (x, y)^{c}$, where the superscripts $\textbf{r}$ and $\textbf{c}$ represent the row and column vectors.

Then by definition of a linear transformation we have: $T(x, y)^{r} + T(u,w)^{r} = (x, y)^{c} + (u, w)^{c} = (x + u, y + w)^{c} = T(x+u, y + w)^{r}$.

Now, you can prove the second condition, namely: $T(\alpha x, \alpha y)^{r} = \alpha T(x, y)^{r}$.