Section 2.18 When the Vectors are the Rows
Until now our ordered n-tuples have appeared in matrices as columns. They can also be placed as rows. The following theorem shows what can be learned by row-reducing such a matrix.
Checkpoint 2.18.1.
Put the points of \(\mathbb{S} = \{ \ot{3}{0}{-3}, \ot{1}{2}{3} \}\) into a matrix as rows then row reduce the resulting matrix. Use \(\mathbb{S}'\) to denote the rows of the resulting matrix. What does Theorem 23 say about \(span\mathbb{S}'\text{?}\)
Checkpoint 2.18.2.
Find a “nicer” set that spans the same space as the set spanned by \(\mathbb{S} = \{ \ot{3}{0}{-3}, \ot{1}{2}{3},\ot{0}{1}{0} \}\text{.}\) Is this new set linearly independent?
Checkpoint 2.18.3.
Find a “nicer” set that spans the same space as the set spanned by \(\mathbb{S} = \{ \ot{3}{0}{-3}, \ot{1}{2}{3}, \ot{3}{2}{1} ,\ot{0}{1}{0} \}\text{.}\) Is this new set linearly independent?
Theorem 2.18.4.
Let \(\mathbb{S}\) be a subset of a linear space \(\mathbb{R}^n\) and let \(\mathbf{P}\) be in \(\mathbb{S}\text{.}\) Assume that \(\mathbf{Q}\) is obtained from \(\mathbf{P}\) either by
multiplying \(\mathbf{P}\) by a non-zero number or
adding to \(\mathbf{P}\) a scalar multiple of another element of \(\mathbb{S}\)
Let \(\mathbb{S}'\) be obtained from \(\mathbb{S}\) by replacing \(\mathbf{P}\) with \(\mathbf{Q}\text{.}\) Then \(\mathbb{S}'\) will span the same linear space as \(\mathbb{S}\text{.}\) In other words: \(span\mathbb{S}' = span\mathbb{S}\text{.}\)
In addition, if \(\mathbb{S}''\) consists of the non-zero rows of a fully simplified matrix, then \(\mathbb{S}''\) is linearly independent.