Question

Suppose V and W are both finite-dimensional. Prove that there exists an injective linear map from V to W if and only if dimV S dimW.

Answer 1

To prove the existence of an injective linear map from V to W if and only if dimV = dimW, we can use the Rank-Nullity Theorem.

Step-by-step explanation:

To prove that there exists an injective linear map from V to W if and only if dimV = dimW, we can use the Rank-Nullity Theorem. According to this theorem, for any linear map T: V -> W, the dimension of the kernel (null space) of T plus the dimension of the image (range) of T is equal to the dimension of V. If T is injective, then the dimension of the kernel is 0, so the dimension of the image must be equal to the dimension of V, i.e., dimV = dimW.