Question

Suppose V and W are finite-dimensional and T ∊ (V,W). Prove that there exist a basis of V and a basis of W such that with respect to these bases, all entries of M(T) are 0 except that the entries in row j, column j, equal 1 for 1 ≤ j ≤ dim range T

Answer 1

To prove the existence of a basis of V and a basis of W such that the entries of M(T) are 0 except for the entries in row j, column j equal to 1 for 1 ≤ j ≤ dim range T, we can utilize the concepts of the range and null space of a linear transformation.

Step-by-step explanation:

To prove that there exist a basis of V and a basis of W such that all entries of M(T) are 0 except for the entries in row j, column j, equal to 1 for 1 ≤ j ≤ dim range T, we can use the concept of the range of a linear transformation. Let's denote the dimension of the range of T as r and take a basis for the range of T, B = {w1, w2, ..., wr}. Then, we can extend this basis to a basis of W, B' = {w1, w2, ..., wr, v1, v2, ..., vn}, where {v1, v2, ..., vn} are vectors that span the null space of T.

Next, we can use the concept of the null space of a linear transformation. Since dim(range T) + dim(null T) = dim(V), we can take a basis for the null space of T, C = {u1, u2, ..., um}. Then, we can extend this basis to a basis of V, C' = {u1, u2, ..., um, x1, x2, ..., xr}, where {x1, x2, ..., xr} are vectors that complete the basis B'.

With respect to the bases C' and B', the matrix M(T) will have all entries as 0, except that the entries in row j, column j, which correspond to the vectors wj in B, will be equal to 1 for 1 ≤ j ≤ dim(range T). Therefore, we have proven the existence of the required bases.