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How many entries are needed to represent a two-dimensional linear transformation as a matrix?

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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.

To represent a two-dimensional linear transformation as a matrix, you need four entries (or coordinates) arranged in a 2-by-2 matrix.1

In two dimensions, a linear transformation is completely determined by its effect on any two vectors that have different directions. Because the transformation of each of these two vectors is defined by two coordinates, a total of four coordinates—two for the first transformed vector and two for the second—are required to specify the transformation.1