How to add and scale linear transformations

We can add the outputs of two transformations that use the same input and output spaces. We can also multiply a transformation’s output by a number. Writing the maps as matrices turns both operations into familiar matrix arithmetic.

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Add the matrices for the two maps

If $S\left(x\right)=Ax$ and $T\left(x\right)=Bx$, then $(S+T)\left(x\right)=(A+B)x$. Use these matrices for $A$ and $B$.

$A=\left[\begin{array}{rr}2& 1\\ -3& 0\end{array}\right]$

$B=\left[\begin{array}{rr}-6& 0\\ 2& -1\end{array}\right]$

Add entries in matching positions to get the matrix of the sum.

$A+B=\left[\begin{array}{rr}-4& 1\\ -1& -1\end{array}\right]$

Multiply every entry to scale a transformation

For $cT$, the matrix is $cB$. With $c=-4$, distribute $-4$ to every entry of $B$.

$-4B=\left[\begin{array}{rr}24& 0\\ -8& 4\end{array}\right]$

Applying this matrix to an input gives $-4$ times the original output $T\left(x\right)$.

Check with a particular input

At $x=(1,2)$, the sum matrix gives $(-2,-3)$. Calculating separately gives $S\left(x\right)=(4,-3)$ and $T\left(x\right)=(-6,0)$. Adding those outputs gives the same $(-2,-3)$.

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