Final answer:
The matrix representation with respect to nonstandard bases is a way of representing linear transformations using matrices when the bases of the vector space are nonstandard.
Step-by-step explanation:
The matrix representation with respect to nonstandard bases is a way of representing linear transformations using matrices when the bases of the vector space are nonstandard. In this representation, the columns of the matrix correspond to the images of the nonstandard basis vectors under the linear transformation.
For example, if we have a linear transformation T: V --> W, where V is a vector space with a nonstandard basis {v1, v2, v3} and W is a vector space with a standard basis {w1, w2, w3}, then the matrix representation of T with respect to the nonstandard bases is a 3x3 matrix where the i-th column is the image of vi under T expressed as a linear combination of the standard basis vectors of W.
By using the matrix representation with respect to nonstandard bases, we can perform computations and analyze the properties of linear transformations in a more convenient and efficient way.