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Skill: Apply a transformation matrix to coordinate points

Explanation and Free Practice Resources

Applying a 2×22\times2 matrix to a coordinate point means multiplying it by the point’s coordinate column vector; the resulting ordered pair gives the image under a linear transformation of the plane. The matrix entries determine how the original coordinates are combined, producing transformations such as rotations, reflections, stretches, and shears; the scope is two-dimensional coordinate transformations, not higher-dimensional or more abstract matrix mappings.

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Detailed Explanation: Apply a transformation matrix to coordinate points

To find the image of a point, write its coordinates as a column vector and multiply the matrix by that vector. Each row of the matrix combines the original coordinates to produce one new coordinate.

For example, find the image of (4,2)(4,2) under the matrix

A=(21−13).A=\begin{pmatrix}2&1\\-1&3\end{pmatrix}.

Write the point as a column vector and multiply:

(21−13)(42)=(2(4)+1(2)−1(4)+3(2))=(102).\begin{pmatrix}2&1\\-1&3\end{pmatrix} \begin{pmatrix}4\\2\end{pmatrix} = \begin{pmatrix}2(4)+1(2)\\-1(4)+3(2)\end{pmatrix} = \begin{pmatrix}10\\2\end{pmatrix}.

So the image of (4,2)(4,2) is the ordered pair (10,2)\boxed{(10,2)}.

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    Matrices - Multiply - Matrix by Vector (as Coordinates) to Values

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