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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 this property is true.Verify that rank $\mathbf{u v}^{T} \leq 1$ if $\mathbf{u}=\left[\begin{array}{r}{2} \\ {-3} \\ {5}\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{l}{a} \\ {b} \\ {c}\end{array}\right]$

$\operatorname{rank} \mathbf{u v}^{T}$

Calculus 3

Chapter 4

Vector Spaces

Section 6

Rank

Vectors

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