Suppose V and W are finite dimensional vector spaces, and and are linear transformations such that for every v in V and for every w in W. If the matrices , represent , respectively (with respect to the same bases for V and W), then .

Question

Suppose V and W are finite dimensional vector spaces, and  and  are linear transformations such that  for every v in V and  for every w in W. If the matrices  ,  represent  ,  respectively (with respect to the same bases for V and W), then  .

Quizplus · Accepted Answer

This is a statement of the Rank-Nullity theorem, which states that for any linear transformation f: V → W, the rank of f plus the nullity of f equals the dimension of V. Here, the statement implies that the rank of T plus the nullity of S equals the dimension of V, which is true.

Suppose V and W Are Finite Dimensional Vector Spaces, and