If the equation $A\vec x = \vec b$ has atleast one solution for each $\vec b$ in $R^n$, then the solution is un ique for each $\vec b$.

True or False?

Question

If the equation $A\vec x = \vec b$ has atleast one solution for each $\vec b$ in $R^n$, then the solution is un ique for each $\vec b$.

True or False?

amistre64 · Answer

say $A=\begin{pmatrix}1&0&3\0&1&4\end{pmatrix} $

wouldnt this suggest that b has at least one solution, and that it is not unique?

amistre64 · Answer

i forgot to read:  "assume A is an nxn matrix ..."  hmm but still it doesnt suggest that its linearly independant does it

kinggeorge · Answer

I think you also have to consider that it has at least one solution for every $\vec b \in \mathbb{R}^n$. And that A is an $n\times n$ matrix...

amistre64 · Answer

2 solutions is "at least" one solution ...

kinggeorge · Answer

In that case, I'm pretty sure it's true. Especially since $\mathbb{R}^n$ is a finite field extension over $\mathbb{R}$.

amistre64 · Answer

that sounds abstract albegra-y

kinggeorge · Answer

On the other hand, what if A doesn't have an inverse. Would that change things?

amistre64 · Answer

So, if an Anxn hits every b in R^n; then it would span R^n .... right?

amistre64 · Answer

if it spans then its linearly independant; has in inverse .. yada yada yada

kinggeorge · Answer

Right. So this statement would be true then.

amistre64 · Answer

yeah, I just dint read all the pertinent information to begin with on the exam :)  it makes more sense now