Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 4: Q4.5-33E (page 191)
[M] According to Theorem 11, a linearly independent set \(\left\{ {{{\bf{v}}_1},....{{\bf{v}}_k}} \right\}\) in \({\mathbb{R}^n}\) can be expanded to a basis for \({\mathbb{R}^n}\). One way to this is to create \(A = \left[ {\begin{array}{*{20}{c}}{{{\bf{v}}_1}}& \cdots &{{{\bf{v}}_k}\,\;\,\begin{array}{*{20}{c}}{{{\bf{e}}_1}}& \cdots &{{{\bf{e}}_n}}\end{array}}\end{array}} \right]\), with \({{\bf{e}}_1}\),鈥�.,\({{\bf{e}}_n}\) the columns of identity matrix; the pivot columns of A form a basis for \({\mathbb{R}^n}\).
a. Use the method described to extend the following vectors to a basis for \({\mathbb{R}^5}\).
\({{\bf{v}}_1} = \left[ {\begin{array}{*{20}{c}}{ - {\bf{9}}}\\{ - {\bf{7}}}\\{\bf{8}}\\{ - {\bf{5}}}\\{\bf{7}}\end{array}} \right]\), \({{\bf{v}}_2} = \left[ {\begin{array}{*{20}{c}}{\bf{9}}\\{\bf{4}}\\{\bf{1}}\\{\bf{6}}\\{ - {\bf{7}}}\end{array}} \right]\), \({{\bf{v}}_3} = \left[ {\begin{array}{*{20}{c}}{\bf{6}}\\{\bf{7}}\\{ - {\bf{8}}}\\{\bf{5}}\\{ - {\bf{7}}}\end{array}} \right]\)
b. Explain why the method works in general: Why are the original vectors \({{\bf{v}}_1}\),鈥�.,\({{\bf{v}}_k}\) included in the basis found for Col A? Why is \({\bf{Col}}\,A = {\mathbb{R}^n}\)?
Short Answer
a. \(\left\{ {{{\bf{v}}_1},\,\,{{\bf{v}}_2},\;\;{{\bf{v}}_3},\;\;{{\bf{e}}_2},\;{{\bf{e}}_3}} \right\}\)
b. \({\rm{Col}}\left( A \right) = {\mathbb{R}^n}\)
Step by step solution
Over 30 million students worldwide already upgrade their learning with 91影视!
