Hi,

How can I show that th set of all solutions to Ax = b where A is mxn, is not a subspace of R^n if b <> 0.

If Ax = 0, (b = 0), the set of solutions IS a subspace. What difference does it make whether b is 0 or not?

Thanks!

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- Nov 2nd 2008, 02:32 PMkpizlesubspaces
Hi,

How can I show that th set of all solutions to Ax = b where A is mxn, is not a subspace of R^n if b <> 0.

If Ax = 0, (b = 0), the set of solutions IS a subspace. What difference does it make whether b is 0 or not?

Thanks! - Nov 3rd 2008, 01:55 PMCaptainBlack
Suppose $\displaystyle x$ is a solution to this and $\displaystyle \lambda$ is a scalar, then $\displaystyle A(\lambda x)=\lambda b$, so $\displaystyle \lambda x$ is not in the set of solutions unless $\displaystyle b=0$, but if the set of solutions is a subspace it would have to be.

CB