Inverse Linear Transformations

Recall that a matrix is a linear transformation. The inverse of a particular linear transformation is said transformation in reverse. That is, if a transformation is applied, and then the inverse of that transformation is applied, the vectors all return to their original location. For example, the inverse of a a clockwise rotation 90 degrees is a counter-clockwise rotation 90 degrees.

Recall that a matrix composition is taking two linear transformations, and turning them into one transformation, such that applying the two transformations, one after the other, is the same as applying the single, composed matrix. This means the result of a applying a transformation and then its inverse is the same as doing nothing, which is the identity matrix.

The linear operator on defined by for all is called the identity operator or identity transformation.

Note that the matrix associated with the linear operator is the identity matrix

If is a linear operator on and there exists another linear operator on such that , then we say that is invertible, and call the inverse of , and write . Also, , , and . In addition, , and .

(see Composition of Linear Transformations, Matrix Inverses, standard matrix).