Determining if a map is linear from vector mapping I know we can say a map is linear if $f(\alpha x+\beta y)=\alpha f(x)+\beta f(y)$ is true.
The issue I'm having is converting the maps to the functions so that I can apply that definition.
$\mathbb{R}^2\ni \begin{bmatrix}x \\ y \end{bmatrix} \rightarrow \begin{bmatrix}x^2 \\ xy \\y^2 \end{bmatrix} \in \mathbb{R}^3$
$\mathbb{R}^3\ni \begin{bmatrix}x \\ y \\z \end{bmatrix} \rightarrow \begin{bmatrix}1+x \\ 1+y \\z \end{bmatrix} \in \mathbb{R}^3$
How do I test if these maps are linear?
EDIT: More specifically, how do I write these mappings as functions so that that I can apply the definition of linearity?
EDIT2: How do I give the matrix expression for the map?
 A: The first one is equivalent to $f(x,y)=(x^2,xy,y^2) $ you can input the values $(ax+w,by+u) $ and check if it is linear (Here $x, w, y$ and $u$ are vectors and $a, b$ are scalars.), the second one is $f(x,y,z)=(1+x,1+y,z) $ you can input the value $(ax+p,by+q,cz+r)$to check if it is linear. (Here $a, b$ and $c$ are scalars and $x, p, y, q, z$ and $r$ are vectors.)
A: Hints:
=== A linear transformation maps the zero vector to the  zero vector
== A linear map is closed under addition.
For example, in the first case, can you check whether
$$\phi\left(\binom10+\binom01\right)\stackrel?=\phi\binom10+\phi\binom01$$
namely: is it true that
$$\phi\binom11=\begin{pmatrix}1\\1\\1\end{pmatrix}\stackrel?=\phi\binom10+\phi\binom01$$
A: If you want to write them as functions, you may write $f\left(\begin{bmatrix}x \\ y \end{bmatrix}\right)=\begin{bmatrix}x^2 \\ xy \\y^2 \end{bmatrix}$ and similar for the other one.
One thing we can say about linear functions into vector spaces is that they will also project to linear functions in each component. Since $f(x,y)=x^2$ is not linear, the first component function of the first function is not linear, so the function is not linear.
