Q40E Let $A$ be an $m \times n$ m... [FREE SOLUTION]

91影视

Linear Algebra and its Applications

David C. Lay, Steven R. Lay and Judi J. McDonald

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 483 pages

ISBN: 978-03219822384

Chapter 1: Q40E (page 1)

Let $A$ be an $m \times n$ matrix, and let $u$ and ${\mathop{\rm v}\nolimits} $ be vectors in ${\mathbb{R}^n}$ with the property that $A{\mathop{\rm u}\nolimits} = 0$ and $A{\mathop{\rm v}\nolimits} = 0$. Explain why $A\left( {u + v} \right)$ must be the zero vector. Then explain why $A\left( {c{\mathop{\rm u}\nolimits} + d{\mathop{\rm v}\nolimits} } \right) = 0$ for each pair of scalars $c$ and $d$.

Short Answer

Expert verified

$A\left( {u + v} \right) = 0,A\left( {cu + dv} \right) = 0$ is proved.

Step by step solution

Determine if $A\left( {u + v} \right)$ is the zero vector

Theorem 5tells that$A$is an$m \times n$matrix;${\mathop{\rm u}\nolimits} $and${\mathop{\rm v}\nolimits} $are vectors in${\mathbb{R}^n}$, and$c$is a scalar, then

$A\left( {u + v} \right) = Au + Av$.

It is given that$u$and$v$are vectors in${\mathbb{R}^n}$with the properties$Au = 0$and$Av = 0$. Use theorem 5 to get

$\begin{array}{c}A\left( {u + v} \right) = Au + Av\\ = 0 + 0\\ = 0.\end{array}$

Determine $A\left( {cu + dv} \right)$ for each pair of scalars

Theorem 5tells that$A$is an$m \times n$matrix;${\mathop{\rm u}\nolimits} $and${\mathop{\rm v}\nolimits} $are vectors in${\mathbb{R}^n}$, and$c$is a scalar, then

a.$A\left( {u + v} \right) = Au + Av$

b.$A\left( {cu} \right) = c\left( {Au} \right)$.

Consider$c$and$d$as the scalars. Use the sub-parts of theorem 5 to get

$\begin{array}{c}A\left( {c{\mathop{\rm u}\nolimits} + d{\mathop{\rm v}\nolimits} } \right) = A\left( {c{\mathop{\rm u}\nolimits} } \right) + A\left( {d{\mathop{\rm v}\nolimits} } \right)\\ = c \cdot Au + d \cdot Av\\ = c0 + d0\\ = 0.\end{array}$

Thus, $A\left( {c{\mathop{\rm u}\nolimits} + d{\mathop{\rm v}\nolimits} } \right) = 0$ is proved.