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How can a linear transformation rotate a geometric figure?

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vector parallelogram for addition and subtraction — linear algebra One method of adding and subtracting vectors is to place their tails together and then supply two more sides to form a parallelogram. The vector from their tails to the opposite corner of the parallelogram is equal to the sum of the original vectors. The vector between their heads (starting from the vector being subtracted) is equal to their difference.

In linear algebra, a linear transformation acts as a function that maps vectors from one position to another while preserving vector addition and scalar multiplication.1

A rotation is a specific type of linear transformation that alters the direction of vectors while keeping their lengths unchanged. To perform this transformation, the operation is represented by a matrix—a rectangular array of scalars. In two dimensions, for example, the transformation is determined by its effect on two vectors with different directions; the resulting coordinates are arranged into a 2-by-2 matrix.1

When applying this transformation to a geometric figure, the matrix acts on the coordinates of every point in the figure, effectively rotating the entire shape around a fixed point, such as the origin. Because these transformations are linear, they maintain the structural relationships between the points of the figure throughout the rotation.21