I need help to prove this in Linear Algebra

If The linear transformation *x *→ *Ax *maps **R***n *onto **R***n*.

then

The columns of *A *span **R***n*.

I proved the other way...but in order for this proof to be complete...I need to prove this..step by step please...it is important that I do understand the proof as well....thanks in advance

Re: I need help to prove this in Linear Algebra

This is similar to the other problem. Given x = (x_{1}, ..., x_{n})^{T}, where the superscript T means transpose, i.e., this is a column vector, Ax = x_{1}A_{1} + ... + x_{n}A_{n} where A_{i} is the *i*th column of A and this is the sum of column vectors. The fact that x ↦ Ax is onto means that for every b ∈ ℝ^{n} there exist an x such that Ax = b. But this also means that the columns of A span ℝ^{n}.

Re: I need help to prove this in Linear Algebra

Doesn't it follow from the fact that the linear transformation is "onto" ?

Means that any vector in $\displaystyle \mathbb{R}^{n} $ can be gotten by doing Ax, where x is the coordinate written under some basis B. That itself means that any vector in $\displaystyle \mathbb{R}^n$ can be written in terms of $\displaystyle c_1 v_1 + ... +c_n v_n $ where $\displaystyle v_1,..v_n$ are the columns of $\displaystyle A$ and $\displaystyle c_1,..c_n$ is a coordinate under some basis B. thus the columns of A span $\displaystyle \mathbb{R}^n$