Section 2.4 Viewing subspaces through different lenses
While it is true that we can create a subspace by taking the span of an arbitrary collection of vectors, sometimes it is nice to have a spanning set that is particularly efficient or has other special properties. So we review how we can manipulate vectors in the row or column space of a matrix to produce nice(r) spanning sets.
You should remind yourself why the following proposition is true.
Proposition 2.4.1.
Let \(A\) be an \(m\times n\) matrix with coefficients in field \(F,\) and let \(R\) be its reduced row-echelon form. Then row space of \(A\) is the same as the row space of \(R\text{.}\) More precisely, elementary row operations on a matrix do not change its rowspace.
It follows that the span of the nonzero rows in the reduced row-echelon form
\(R\) is a minimal spanning set for the row space, better known as a
basis.
Keeping with
\(A\) an
\(m\times n\) matrix with coefficients in field
\(F,\) another subspace is called the
nullspace of
\(A\) (which is the same as the
kernel of the linear transformation which takes
\(x \mapsto Ax\)), that is
\begin{equation*}
\{x \in F^n \mid Ax = \0\}.
\end{equation*}
In your course, you also proved:
Proposition 2.4.2.
The nullspace of \(A\) is the same as the nullspace of \(R= RREF(A),\) since the set of solutions to \(Ax=\0\) is exactly the same as the set of solutions to \(Rx=\0.\)
