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How many entries are needed to represent a three-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 three-dimensional linear transformation as a matrix, you need 9 entries.

In linear algebra, a linear transformation in three dimensions is determined by its effect on three vectors. Since each of these three transformed vectors requires three coordinates to be specified, the resulting arrangement is a 3-by-3 matrix, which contains a total of $3 \times 3 = 9$ entries.1