I started by making a basis of U and a basis of V.
Transforming the basis of U by T gives . These vectors may not be linearly independent so sifting them gives:
where . Let which gives me the basis of V.
Is this the correct way to start? I really can't see another way of doing this question.
EDIT: Sorry for mistyping "Dimension" as the title.
Let and be a basis for and be a basis for . This means that . Let be the matrix i.e. is the matrix associated with . We see that and . Therefore, defined by is a linear transformation between the vector space and - the matrices over . This linear transformation is one-to-one because if it means and so for each . All elements in are linear combinations of and so if then it means is the zero linear transformation. Thus, and that implies is one-to-one. It remains to show that is onto. Thus, provides a vector space isomorphism between and . Now, has dimension with the basis where is a matrix having in location and 's everywhere else. Thus, .Let U and V be vector spaces of dimensions n and m over K, and let be the vector space over K of all linear maps from U to V.
Find the dimension and describe a basis of .