Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
We have solutions for your book!
See our solution for Question 9E from Chapter 4.1 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 9E
Chapter:
Problem:
Let H be the set of all vectors of the form...
Step-by-Step Solution
Step 1
Given set of vectors\[H = \left[ {\begin{array}{*{20}{c}}s\\{3s}\\{2s}\end{array}} \right]\]We have to find a vector v in $R^3$ such that\[H = {\rm{Span}}\left( {\bf{v}} \right)\]
Step 2: The required vector
\[\begin{array}{l}H = \left[ {\begin{array}{*{20}{c}}s\\{3s}\\{2s}\end{array}} \right]\\ = s\left[ {\begin{array}{*{20}{c}}1\\3\\2\end{array}} \right]\end{array}\]From the above result we see that H spans in subspace of vector $\bf{v}$ where $\bf{v}$ is:\[{\bf{v}} = \left[ {\begin{array}{*{20}{c}}1\\3\\2\end{array}} \right]\]
Step 3
From Theorem-1: If ${{\bf{v}}_1},\,{{\bf{v}}_2}...{{\bf{v}}_p}$ are in a vector space V , then ${\rm{Span}}\left( {{{\bf{v}}_1},\,{{\bf{v}}_2}...{{\bf{v}}_p}} \right)$ is a subspace of V . hence the set H is a subspace of $R^3$
ANSWER
\[{{\bf{v}}_1} = \left[ {\begin{array}{*{20}{c}}1\\3\\2\end{array}} \right]\]
Given set of vectors\[H = \left[ {\begin{array}{*{20}{c}}s\\{3s}\\{2s}\end{array}} \right]\]We have to find a vector v in $R^3$ such that\[H = {\rm{Span}}\left( {\bf{v}} \right)\]
Step 2: The required vector
\[\begin{array}{l}H = \left[ {\begin{array}{*{20}{c}}s\\{3s}\\{2s}\end{array}} \right]\\ = s\left[ {\begin{array}{*{20}{c}}1\\3\\2\end{array}} \right]\end{array}\]From the above result we see that H spans in subspace of vector $\bf{v}$ where $\bf{v}$ is:\[{\bf{v}} = \left[ {\begin{array}{*{20}{c}}1\\3\\2\end{array}} \right]\]
Step 3
From Theorem-1: If ${{\bf{v}}_1},\,{{\bf{v}}_2}...{{\bf{v}}_p}$ are in a vector space V , then ${\rm{Span}}\left( {{{\bf{v}}_1},\,{{\bf{v}}_2}...{{\bf{v}}_p}} \right)$ is a subspace of V . hence the set H is a subspace of $R^3$
ANSWER
\[{{\bf{v}}_1} = \left[ {\begin{array}{*{20}{c}}1\\3\\2\end{array}} \right]\]