Let

A ∈ Mn(F).

Prove thatA ∈GLn(F) if and only if the columns ofAform a basis for Fn.

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- Nov 3rd 2009, 09:38 PMamm345General Linear Group Proof Help
Let

*A ∈ Mn*(F).

Prove that*A ∈*GL*n*(F) if and only if the columns of*A*form a basis for F*n*.

- Nov 4th 2009, 01:55 AMHallsofIvy
$\displaystyle GL_n(F)$ is the group of

**invertible**n by n matrices on F. The columns of a matrix span the image of the matrix in $\displaystyle F^n$. If those columns form a basis for $\displaystyle F^n$, then the image is the entire space and so the matrix is invertible.