Question

Suppose V and W are finite-dimensional, and T is in L(V;W). Prove that there exists 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(rangeT).
Options:
a) Basis transformation and matrix manipulation
b) Null matrix construction
c) Range mapping and basis transformation
d) Identity matrix construction

Answer 1

Basis transformation and matrix manipulation. Hence the correct option is a.

Basis Transformation:

Since V and W are finite-dimensional, we can choose bases for both V and W.

Let {v1 ,v2​,…,vn​} be a basis forV, and BW={w1 ,w2 ,…,wm } be a basis for W.

Matrix Representation (M(T)):

The matrix M(T) is the matrix representation of the linear operator T with respect to the chosen bases.

The entries in M(T) correspond to the coefficients of the linear combination of the basis vectors.

Construction of M(T):

For each basis vector ∈B​V corresponding to a vector in the range of T, set the j-th column of M(T) to have 1 in the j-th row and 0 in all other rows.

This ensures that the linear transformation T maps each basis vector in V to the corresponding basis vector in W that lies in the range of T.

Zero Entries:

For all other columns in M(T), set the entries to be 0, ensuring that the linear transformation does not affect vectors outside the range of T.

This construction guarantees that with respect to the chosen bases, the matrix M(T) has the desired properties. Hence the correct option is a.