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Prove that β is a basis for Fn if and only if det(B)≠0.

For one direction of the proof I discussed this with a peer:

Since β consists of n vectors, β is a basis if and only if these vectors are linearly independent, which is equivalent to the map L_B being one-to-one. Since the matrix B is square, this is in turn equivalent to B being invertible, hence having a nonzero determinant.

However I do not understand the transition from the vectors being linearly independent to being one to one. Why is this true? Also, how do I prove the reverse direction?