Theorem

Suppose V and W are finite-dimensional vector spaces such that dim V > dim W. Then no linear map from V to W is injective.

Proof

Let T be a linear map from V to W. Then ${latex.inline\text{dim null T} = \text{dim V} - \text{dim range T} \geq \text{dim V} - \text{dim W} \gt 0}. Where the first equality comes from 1756253933 - Axler 3.21 Fundamental theorem of linear maps.|3.21, the second line follows from 1753403591 - Axler 2.37 Dimension of a subspace| 2.37. This states that the dimension of null T is greater than 0, which by 1756253705 - Axler 3.15 A function is injective if and only if the null space is {0}|3.15 means T is not injective.