Definiton: The column space of an $m \times n$ matrix $A$, written as $\operatorname{Col} A$, is the set of all linear combinations of the columns of $A$. If $A = [ a_1 \ldots a_n ]$, then $\operatorname{Col} A = \operatorname{Span}\{a_1,\ldots,a_n\}$.
$$A= \begin{bmatrix} 2 & 4 & -2 & 1 \\ -2 & -5 & 7 & 3 \\ 3 & 7 & -8 & 6
\end{bmatrix}$$Find a nonzero vector in $\operatorname{Col} A$.
Solution: It is easy to find a vector in $\operatorname{Col} A$. Any column of $A$
will do.
I'm confused on why any column of $A$ works because $A$ in rref form is:
$$\begin{bmatrix} 1 & 0 & 9 & 0 \\ 0 & 1 & -5 & 0 \\ 0 & 0 & 0 & 1
\end{bmatrix}$$
So wouldn't only columns 1, 2, and 4 work because column 3 is linearly dependent, and thus not a part of the span?
Best Answer
The third column is also in the span because it can be written as $$Col 3=0.Col1+0.Col2+1.Col3+0.Col4$$ What the rref tells you is that the columns $1, 2$ and $4$ not only span $Col A$ but they are also linearly independent. Hence $Col 3$ could alternately be written as $$Col3=a.Col1+b.Col2+c.Col4$$ for some $a, b, c$.