Representing a function in vector form. (2)

Question

First see this previous question: Representing a function in vector form.

.

Can we re-write the following linear function

$$f(w, x, y) = aw + bx + cy + d$$

in the following form as a vector

$$f(w, x, y) = a \vec i + b \vec j + c \vec k + d$$

?

If not, what is the correct form if we consider $w,x,y$ as vectors?

Note: Arrow represents a vector.

Answer 1

If you want your function to be scalar valued, then the answer given to your previous post looks good. Typically, in linear algebra, you have an equation like $y(x) = A \vec{x}$, where $x$ is a vector. Here, you can look at this like $$y(x) = (A_1 \cdot x)e_1 + (A_2 \cdot x)e_2 + ... + (A_n \cdot x)e_n,$$ where $e_i,1 \leq i \leq n$ are the coordinate vectors (like $i,j,k$,etc) of $x$ and $A_i$ are the rows of matrix $A$. Each row of $A$ is a vector with the same coordinate vectors as $x$. This is an example of a vector-valued function. So, $$f(x,w,y) = ax+bw+cy$$ makes sense as a vector valued function, but $$f(x,w,y) = ax+bw+cy+d$$ makes sense only if $d$ is also a vector.