Linear Algebra I · Linear maps, kernels, images, and rank-nullity
Equal dimensions tie injectivity to surjectivity
Problem
Let $K$ be a division ring, let $V,W$ be finite-dimensional left $K$-vector spaces with $\dim_K V=\dim_K W$, and let $T:V\to W$ be linear. Prove that $T$ is injective if and only if $T$ is surjective.
Hint
Translate injectivity into zero nullity and compute the rank.
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