Let A be an n x n matrix. Denote it's columns b v_{1}, v_{2}, ... v_{n}. Keep in mind these are column vectors in R^{n}.

Prove that A is invertible if and only if {v_{1}, v_{2}, ... v_{n}} spans R^{n}.

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- Apr 25th 2013, 04:12 PMwidenerl194proof using invertible and spanning
Let A be an n x n matrix. Denote it's columns b v

_{1}, v_{2}, ... v_{n}. Keep in mind these are column vectors in R^{n}.

Prove that A is invertible if and only if {v_{1}, v_{2}, ... v_{n}} spans R^{n}. - Apr 25th 2013, 05:24 PMGusbobRe: proof using invertible and spanning
I have no idea what you know and can use. But here is a quick proof using things I feel ought to be done before proving this theorem.

spans

iff

is consistent for all

iff

has pivot rows (and also pivot columns since is square)

iff

is row equivalent to the identity matrix

iff

is invertible. - Apr 25th 2013, 05:26 PMHallsofIvyRe: proof using invertible and spanning
First show that if you take A times you get the first column of A, that if you take A times you get the second column, etc.

In other words, A maps the standard basis for R^{n}into the columns of A. Those vectors span the range of A.