Consider a finite-dimensional vector space $V$ with $\dim{V}=n\ge2$.
Consider two subspaces $S_1$ and $S_2$.
Additionally, let $A$ be the matrix formed by basis vectors of $S_1$ acting as columns of $A$. Similarly, let $B$ be the matrix formed by basis vectors of $S_2$ acting as columns of $B$.
According to my understanding (knowledge),
- $A\overset{r}\equiv B\iff N(A)=N(B)$.
- RRE form of $A=$ RRE form of $B\iff S_1=S_2$, i.e., $S1$ and $S_2$ span the same subspace.
My main doubt is, under what conditions statement $1\implies$ statement $2$ ?
and under what conditions statement $2\implies$ statement $1$ ? Please show counter-examples, wherever necessary.
Also tell, how do we differentiate between $S_1$ and $S_2$ when $S_1\neq S_2$ but $\dim{S_1}=\dim{S_2}$. (Which property like rank/nullity or something else make them different? )
The premise here is that there exist subspaces having same dimension, but different set of vectors.
Remark: $N(A)$ denotes null-space of matrix $A$ and
$A\overset{r}\equiv B$ means $A$ is row-equivalent to $B$.
Take this with a grain of salt as Linear Algebra isn't my forte, but I think statement 1 and 2 are equivalent, as A and B's rowspaces span the same subspace iff they have the same RRE