I have a question about determining a basis of $\mathrm{Col(A)}$.
If a matrix $A$ is given, I understand that I have to row reduce matrix $A$ into a reduced echelon form (call this $U$). By looking at the pivots, I can determine which columns of matrix $A$, must be in the span of $\mathrm{Col(A)}$.
But what if I don't get the matrix $A$ given, but only a matrix $B$ which is row equivalent to $A$?
Then there are two options. First row reduce the matrix $B$ and look at the pivot columns of the reduced echelon form of $B$.
If $\mathrm{Col(A)}$ has dimension $2$ for example in $\Bbb R^3$ (i.e. $\Bbb R^m$ of matrix of $m \times n$), then we can't determine $\mathrm{Col(A)}$? Is this true? And why is this true? And what if matrix $A$ is given in this situation? May we then still take 2 columns of $A$ in $\Bbb R^3$ as $\mathrm{Col(A)}$?
If $\mathrm{Col(A)}$ has dimension, that is the number of pivots, $m$ in $\Bbb R^m$ (with $m$ is the number of rows of matrix $A$), then we may take the columns of the reduced echelon form of $B$? Is this true? Or can you only do this if these are vectors of the identity matrix for example $[1,0,0]$? So is this not the case for another vector/column?
So, summarizing, when may you take the columns of the reduced echelon form of $A$ for $\mathrm{Col(A)}$? Is it only if the reduced echelon form consist of vectors of the identity matrix or if $\mathrm{Col(A)}$ has dimension $m$ in $\Bbb R^m$ or both?
I am a beginner in linear algebra, so I would be really thankful if someone could explain this to me.