Algebra1

Let A be an m x n matrix, show that the following are equivalent:
(a) A has a right inverse, that is, there exists an n x m matrix C such that AC=I.
(b) The system Ax=b has at least one solution x for each b in Rm.
(c) The columns of A span Rm.

Show that any one of (a),(b) or (c) implies that m<=n.


Thanks very much.

Quote:

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Originally Posted by suedenation

Let A be an m x n matrix, show that the following are equivalent:
(a) A has a right inverse, that is, there exists an n x m matrix C such that AC=I.
(b) The system Ax=b has at least one solution x for each b in Rm.
(c) The columns of A span Rm.

Show that any one of (a),(b) or (c) implies that m<=n.


Thanks very much.

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This is a one of those proof that is 'natsy' to write out and relatively easy. It is partically a definition of what vector spaces are. Anywars this is called the "Fundamental Theorem of Existence of a Solution for a Linear System" (long name), maybe, if you want write out for you?