Determining whether a matrix function is linear?

[Crosshash]

1. The problem statement, all variables and given/known data
Which of the following functions is linear? Give reasons if they are not linear. If they are linear, give the corresponding matrix.


2. Relevant equations

a)
R
\begin{pmatrix}
x\\
y\\
\end{pmatrix} =

\begin{pmatrix}
x - y \\
x\\
\end{pmatrix}


b)
S
\begin{pmatrix}
a\\
b\\
\end{pmatrix} =

\begin{pmatrix}
5 \\
b\\
\end{pmatrix}


c) T(x \textbf{i} + y \textbf{j} + z \textbf{k}) = 6x \textbf{i} + y \textbf{j} - y \textbf{k}

d)
U
\begin{pmatrix}
x\\
y\\
z\\
\end{pmatrix} =

\begin{pmatrix}
xy \\
y\\
z\\
\end{pmatrix}


3. The attempt at a solution

I have no idea how to do this question, I've tried to find some advice online and neither my notes nor the resources seem to help me.

Could someone possibly tell me what i'm supposed to do please?

Thanks

[VeeEight]

If a function f is linear then f (x+y) = f(x) + f(y) and f(cx) = cf(x) for a constant c and x, y in your domain. Check to see if the functions satisify these two criteria by checking with arbitrary scalar c and variables x and y.

[Crosshash]

Ok, I think I understand, so if I take the first question and use the criteria f(cx) = cf(x).

a)

\begin{pmatrix}
a & b\\
c & d\\
\end{pmatrix}
\begin{pmatrix}
x\\
y\\
\end{pmatrix} = \begin{pmatrix}
x - y \\
x\\
\end{pmatrix}


I get


\begin{pmatrix}
ax + by\\
cx + dy\\
\end{pmatrix} = \begin{pmatrix}
x - y \\
x\\
\end{pmatrix}



So R = \begin{pmatrix}
1 & -1\\
1 & 0\\
\end{pmatrix}


So my R value is this constant? So now I just need to multiply f(x) by R and see if I still get the answer right?

[VeeEight]

No, you do not assume that f(cx) = cf(x). You want to check if that is true. So compute cf(x) and compute f(cx) and see if they are equal. Then move onto computing f(x+y) and f(x) + f(y) and see if they are equal. The function is linear if and only if those two statements are true. If one or both are false then the function is not linear.