The identity function is a linear operator when applied to vector spaces. By definition, the identity function from Rn to Rn is the function that each vector takes for itself. It is clear that the identity function is a linear operator whose standard matrix is the identity matrix. We are going to denote the identity operator by ID.

If T is a linear transformation from V to W and k is a scalar, then the kT map, which takes each vector A from V to k times T (A), is again a linear transformation from V to W. A function from Rn to Rm that takes each vector n v to the vector m Av, where A is a matrix of m by n, is called a linear transformation. It can be demonstrated that, if a transformation is defined by formulas in the coordinates as in the previous example, then the transformation is linear if and only if each coordinate is a linear expression in the variables without a constant term. A function T from X to X is called invertible if there is another function S from X to X such that TS%3Dst%3DID, the identity function (that is, if T leads x to y, S must carry y to x).

On the other hand, if BA%3DI, then the linear operator S with standard matrix B is the inverse of T because ST is the linear operator whose standard matrix is I. Therefore, the product ST is a linear transformation and the standard matrix ST is the product of the standard matrices BA. If k is a number and T is a linear transformation from Rm to Rn, then kT is a function from Rm to Rn that takes each vector V from Rm to kT (V). Suppose that T is a linear transformation from Rm to Rn with the standard matrix A and S is a linear transformation from Rn to Rk with the standard matrix B.

By definition, S undoes what T does, that is, if T leads V to W, S must bring W to V (otherwise, ST would not be the identity operator). If T and S are linear transformations from Rm to Rn, then T+S is again a linear transformation from Rm to Rn and the standard matrix of this transformation is equal to the sum of the standard matrices of T and S. If T is a linear transformation from Rm to Rn and k is a scalar, then kT is again a linear transformation from Rm to Rn and the standard matrix of this transformation is equal to k times the standard matrix of T. The product ST of a linear transformation T from Rm to Rn and a linear transformation S from Rn to Rk is a linear transformation from Rm to Rk and the linear transformation from Rm to Rk is a linear transformation from Rm to Rk and the linear transformation from Rm to Rk is a linear transformation from Rm to Rk and the The standard matrix of ST is equal to the product of the standard matrices of S and T.

If T is a linear transformation from V to W and S is a linear transformation from W to Y (V, W, Y are vector spaces), then the product (composition) ST is a linear transformation from V to Y.

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