91Ó°ÊÓ

Exercises \(27-29\) concern an \(m \times n\) matrix \(A\) and what are often called the fundamental subspaces determined by \(A .\) Justify the following equalities: a. \(\operatorname{dim} \mathrm{Row} A+\operatorname{dim} \mathrm{Nul} A=n\) Number of columns of \(A\) b. \(\operatorname{dim} \operatorname{Col} A+\operatorname{dim} \mathrm{Nul} A^{T}=m\) Number of rows of \(A\)

Rank 1 matrices are important in some computer algorithms and several theoretical contexts, including the singular value decomposition in Chapter \(7 .\) It can be shown that an \(m \times n\) matrix \(A\) has rank 1 if and only if it is an outer product; that is, \(A=\mathbf{u v}^{T}\) for some \(\mathbf{u}\) in \(\mathbb{R}^{m}\) and \(\mathbf{v}\) in \(\mathbb{R}^{n} .\) Exercises \(31-33\) suggest why this property is true. Let \(A\) be any \(2 \times 3\) matrix such that rank \(A=1,\) let \(\mathbf{u}\) be the first column of \(A,\) and suppose \(\mathbf{u} \neq \mathbf{0}\) . Explain why there is a vector \(\mathbf{v}\) in \(\mathbb{R}^{3}\) such that \(A=\mathbf{u} \mathbf{v}^{T} .\) How could this construction be modified if the first column of \(A\) were zero?

\([\mathbf{M}]\) Let \(H=\operatorname{Span}\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\\}\) and \(\mathcal{B}=\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\\} .\) Show that \(\mathbf{x}\) is in \(H\) and find the \(\mathcal{B}\) -coordinate vector of \(\mathbf{x},\) for $$ \mathbf{v}_{1}=\left[\begin{array}{r}{11} \\ {-5} \\ {10} \\\ {7}\end{array}\right], \mathbf{v}_{2}=\left[\begin{array}{c}{14} \\ {-8} \\\ {13} \\ {10}\end{array}\right], \mathbf{x}=\left[\begin{array}{r}{19} \\\ {-13} \\ {18} \\ {15}\end{array}\right] $$