Hi all,

I need to prove this:

If The columns of A span Rn, then The equationAx=bhas at least one solution for eachbinRnthanks

Step by step please....i really want to understand this...thanks

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- Mar 19th 2013, 12:11 PMTrystanandEmma1Linear Algebra Proof
Hi all,

I need to prove this:

If The columns of A span Rn, then The equation*Ax*=*b*has at least one solution for each*b*in**R***n*thanks

Step by step please....i really want to understand this...thanks - Mar 19th 2013, 01:12 PMemakarovRe: Linear Algebra Proof
If the columns of A span ℝ

^{n}, it means that for every b ∈ ℝ^{n}there exist numbers x_{1}, ..., x_{n}such that x_{1}A_{1}+ ... + x_{n}A_{n}= b, where A_{i}is the*i*th column of A. But this exactly means that Ax = b has the solution x = (x_{1}, ..., x_{n}). - Mar 19th 2013, 01:30 PMHallsofIvyRe: Linear Algebra Proof
That's pretty much what

**spans**means isn't it? $\displaystyle x_1A_1+ x_2A_2+ \cdot\cdot\cdot+ x_nA_n$ is exactly the definition of "Ax". So that is simply saying, "for any $\displaystyle b\in R^$, there exist x such that Ax= b".