Theorem

Suppose V and W are finite-dimensional vector spaces such that dim V < dim W. Then on linear map from V to W is surjective.

Proof

Let T be a linear map from V to W. Then ${latex.inline\text{dim range T} = \text{dim V} - \text{dim null T} \leq \text{dim V} \lt \text{dim W}}. The equality comes from 1756253933 - Axler 3.21 Fundamental theorem of linear maps.|3.21. The iequality above states that dimesnion of the range is less than dimension of W, which means that the linear map T cannot be surjective.