How to write a linear transformation as a matrix

A linear transformation respects vector addition and scalar multiplication. That means its action on the coordinate vectors tells us how it acts on every vector. Let’s use those images to build a matrix for a reflection and a stretch.

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Put the coordinate-vector images in the columns

For a linear map from the plane to the plane, put $T(1,0)$ in the first column and $T(0,1)$ in the second. Since $(x,y)=x(1,0)+y(0,1)$, the matrix then gives $T(x,y)=xT(1,0)+yT(0,1)$.

Combine a horizontal stretch with a reflection

A horizontal stretch by $3$ changes $x$ to $3x$. A reflection over the horizontal axis changes $y$ to $-y$. Together, the transformation is $T(x,y)=(3x,-y)$.

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

Transform the triangle vertices

Apply the same matrix to each vertex

Original vertex	Image
$(-1,4)$	$(-3,-4)$
$(-3,-2)$	$(-9,2)$
$(4,0)$	$(12,0)$

Connecting the images in the original order gives the transformed triangle. A matrix with m rows and n columns represents a linear map taking n-component inputs to m-component outputs.

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