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 Rn
Step by step please....i really want to understand this...thanksthanks
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 ith column of A. But this exactly means that Ax = b has the solution x = (x_{1}, ..., x_{n}).
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".