# Prove that T is onto if and only if $T^t$ is one-to-one

Let $V$ and $W$ be finite-dimensional vector spaces over a field $F$, and let $T:V → W$ be a linear map. Prove that $T$ is onto if and only if the transpose of $T$ is one-to-one

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It would be better if you didn't state your problem as a demand --- it sounds like you are assigning homework to us. Tell us how you came across the problem, why it's important to you, what you know about it, what progress you have made on it, and so on. –  Gerry Myerson Nov 18 '13 at 7:56

If $\alpha,\beta\in W^*$ then $T^t(\alpha)=T^t(\beta)$ means that $\alpha\circ T=\beta\circ T$, which happens if and only if $\alpha$ and $\beta$ take the same values on the image of $T$. You can argue that this condition implies $\alpha=\beta$ (in other words that $T^t$ is injective) if and only if $T$ is surjective; the "if" part is obvious, and for the "only if" part you must show that in case $T$ fails to be surjective, you can choose $\alpha,\beta$ to coincide in the image of $T$ but not everywhere. For the latter, choose a basis of $W$ that extends a basis of the image of $T$.

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I'm not keen on providing answers to posters who do so little for themselves. –  Gerry Myerson Nov 19 '13 at 9:22
@GerryMyerson: Neither am I. But I tried to show the correct abstract structure of the answer once one has sufficient maturity to understand the notions in this manner, while leaving some easy details to be filled in as well. I think that if OP had this level of understanding, there would have been no need to ask the question; ergo, I don't think I am providing (useful) help with cheating on homework. –  Marc van Leeuwen Nov 19 '13 at 11:23
Fix bases $\mathcal{A}$ and $\mathcal{B}$ of $V$ and $W$. The matrix of $T^*$ (transpose of $T$) relative to $\mathcal{B}^*$ and $\mathcal{A}^*$ (dual bases) is the transpose $A^t$ of the matrix $A$ of $T$ relative to $\mathcal{A}$ and $\mathcal{B}$.
A matrix $A$ is right invertible (i.e., the associated linear map is onto) if and only if its transpose is left invertible (i.e., the associated linear map is one-to-one).