Representing a function in vector form.

Question

As far as I understand, according to linear algebra, linear functions, both single and multivariable, can be represented in vector form.

For instance,

$$z = aw + bx + cy + d$$

can be rewritten as

$$z = \begin{bmatrix}a & b&c &|& d\end{bmatrix} \cdot \begin{bmatrix}w \\ x \\ y \\ 1\end{bmatrix}$$

My questions are:

1. Am I correct with the above?
2. Is it possible to rewrite the function $\bbox[lightgray] {z = w^2 + x^2 + y^2 + 8}$ in vector form?

Answer 1

In the same vein as your example, one can write $$w^2+x^2+y^2+8=\begin{bmatrix}w&x&y&1\end{bmatrix}\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&8\end{bmatrix}\begin{bmatrix}w\\x\\y\\1\end{bmatrix}.$$

Answer 2

If you let $$ u=<x,y,w>$$ Then $$z=u.u+8=||u||^2 +8$$