Linear Alegebra linear equations proofs

• Jan 13th 2010, 05:58 AM
Linear Alegebra linear equations proofs
Prove the following:

A system of linear equations
Ax = b with m < n either has no solution or has several different solutions.

A system of linear equations
Ax = b where A has rank m always has a solution.

I think I'm making a little bit of headway but some help would be much appreciated
• Jan 13th 2010, 07:50 AM
Drexel28
Quote:

Prove the following:

A system of linear equations
Ax = b with m < n either has no solution or has several different solutions.

A system of linear equations
Ax = b where A has rank m always has a solution.

I think I'm making a little bit of headway but some help would be much appreciated

You need to define thoroughly what everything means here.
• Jan 13th 2010, 09:47 AM
Dinkydoe
Suppose A has rank n, that is A is invertable. Then $Ax = b$ has a unique solution $x = A^{-1}b$ by uniqueness of the matrix-inverse.

If A has rank m < n then it is row-equivalent with a matrix with at least one row zero's, say row i, $1 \leq i\leq n$. Suppose it does have a solution $x^* = (x_1,\cdots x_i,\cdots x_n)$. Then for any choice of $x_i\in \mathbb{R}$ we have that $x^*$ is a solution.