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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 thisproperty 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?

$\mathbf{v}^{T}=\left[\begin{array}{lll}{0} & {0} & {1}\end{array}\right]$

Calculus 3

Chapter 4

Vector Spaces

Section 6

Rank

Vectors

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