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Finding the transformation matrix

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Finding the Transformation Matrix

A transformation matrix maps each point of a figure to a new position, rotating, reflecting, or scaling it. Learn how to find the transformation matrix from the images of the basis vectors (1, 0) and (0, 1), which become its columns, and how to apply it to a shape using matrix multiplication.

What a transformation matrix is

A transformation matrix is a matrix that maps every point of a figure to a new position — rotating, reflecting, scaling, or shearing it. Multiplying the matrix by a point's coordinate vector gives the transformed point, so one matrix describes the whole transformation. Applying it uses matrix multiplication.

To find the transformation matrix, look at where it sends the basis vectors (1, 0) and (0, 1). Those two images become the columns of the matrix. Writing it down uses standard matrix notation.

A transformation matrix reflecting a triangle A coordinate grid. The original triangle on the right of the y-axis has vertices (1,1), (4,1), (1,3). The transformation matrix with first row minus one, zero and second row zero, one reflects it across the y-axis to the triangle on the left with vertices (minus 1,1), (minus 4,1), (minus 1,3). x y original image T = [ −1 0 ; 0 1 ] maps (x, y) → (−x, y)
The matrix with rows (−1, 0) and (0, 1) reflects a triangle across the y-axis.

A reflection across the y-axis sends (1, 0) → (−1, 0) and (0, 1) → (0, 1). Those images are the columns, so the matrix has first row (−1, 0) and second row (0, 1). Multiplying it by any point (x, y) returns (−x, y).

The matrix that leaves every point unchanged is the identity matrix, with ones on the diagonal — the transformation that does nothing.

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