Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 4E from Chapter 4.4 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 4E
Chapter:
Problem:
Find the vector × determined by the given coordinate vector and the given basis .
Step-by-Step Solution
Given Information
We have to find the vector x determined by the given coordinate vector ${\left[ x \right]_B}$ and the given basis B. \[ B = \left\{ \left[ \begin{array} { l } { 4 } \\ { 5 } \end{array} \right] , \left[ \begin{array} { l } { 6 } \\ { 7 } \end{array} \right] \right\} , [ { x } ] _ { B } = \left[ \begin{array} { c } { 8 } \\ { - 5 } \end{array} \right] \]
Step-1: The vector x
The vector x can be found by product of basis and given coordinate vector \[ \begin{aligned} { x } & = 8 \left[ \begin{array} { l } { 4 } \\ { 5 } \end{array} \right] - 5 \left[ \begin{array} { l } { 6 } \\ { 7 } \end{array} \right] \\ & = \left[ \begin{array} { c } { 32 } \\ { 40 } \end{array} \right] - \left[ \begin{array} { c } { 30 } \\ { 35 } \end{array} \right] \\ & = \left[ \begin{array} { l } { 2 } \\ { 5 } \end{array} \right] \end{aligned} \] Therefore the required vector is,
The basis for the H is \[{\bf{x}} = \left[ {\begin{array}{*{20}{l}} 2\\ 5 \end{array}} \right]\]
We have to find the vector x determined by the given coordinate vector ${\left[ x \right]_B}$ and the given basis B. \[ B = \left\{ \left[ \begin{array} { l } { 4 } \\ { 5 } \end{array} \right] , \left[ \begin{array} { l } { 6 } \\ { 7 } \end{array} \right] \right\} , [ { x } ] _ { B } = \left[ \begin{array} { c } { 8 } \\ { - 5 } \end{array} \right] \]
Step-1: The vector x
The vector x can be found by product of basis and given coordinate vector \[ \begin{aligned} { x } & = 8 \left[ \begin{array} { l } { 4 } \\ { 5 } \end{array} \right] - 5 \left[ \begin{array} { l } { 6 } \\ { 7 } \end{array} \right] \\ & = \left[ \begin{array} { c } { 32 } \\ { 40 } \end{array} \right] - \left[ \begin{array} { c } { 30 } \\ { 35 } \end{array} \right] \\ & = \left[ \begin{array} { l } { 2 } \\ { 5 } \end{array} \right] \end{aligned} \] Therefore the required vector is,
The basis for the H is \[{\bf{x}} = \left[ {\begin{array}{*{20}{l}} 2\\ 5 \end{array}} \right]\]