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See our solution for Question 1E from Chapter 6.7 from Lay's Linear Algebra and Its Applications, 5th Edition.
Given Information
We are given with following vectors \[ x = ( 1,1 ) \text { and } y = ( 5 , - 1 ) \] We have to find following entities \[ \| \mathbf { x } \| , \| \mathbf { y } \| , \text { and } | \langle \mathbf { x } , \mathbf { y } \rangle | ^ { 2 } \] Then, we have to describe the vectors that are orthogonal to $y$.
Step-1: (a)
Length of vector $x$ in $R^2$: \[ \begin{aligned} \| \mathbf { x } \| & = \sqrt { \langle \mathbf { x } , \mathbf { x } \rangle } \\ & = \sqrt { \langle ( 1,1 ) , ( 1,1 ) \rangle } \\ & = \sqrt { 4 ( 1 ) ( 1 ) + 5 ( 1 ) ( 1 ) } \\ & = \sqrt { 9 } \\ = { 3 } \end{aligned} \] Length of vector $y$ in $R^2$: \[ \begin{aligned} \| \mathbf { y } \| & = \sqrt { \langle \mathbf { y } , \mathbf { y } \rangle } \\ & = \sqrt { \langle ( 5 , - 1 )
, ( 5 , - 1 ) \rangle } \\ & = \sqrt { 4 ( 5 ) ( 5 ) + 5 ( - 1 ) ( - 1 ) } \\ & = \sqrt { 105 } \end{aligned} \] And, $| \langle \mathbf { x } , \mathbf { y } \rangle | ^ { 2 }$ \[\begin{array}{l} |\langle x,y\rangle {|^2} = |\langle (1,1),(5, - 1)\rangle {|^2}\\ = |4(1)(5) + 5(1)( - 1){|^2}\\ = |15{|^2}\\ = 225 \end{array}\]
Step-2: (b)
We have to describe all vectors $\left( z _ { 1 } , z _ { 2 } \right)$ that are orthogonal to $y$. A vector is orthogonal
to y if and only if $\langle \mathbf { z } , \mathbf { y } \rangle = 0$ \[ \begin{aligned} \langle \mathbf { z } , \mathbf { y } \rangle & = 0 \\ \left\langle \left( z _ { 1 } , z _ { 2 } \right) , ( 5 , - 1 ) \right\rangle & = 0 \\ 4 \left( z _ { 1 } \right) ( 5 ) + 5 ( - 1 ) \left( z _ { 2 } \right) & = 0 \\ 20 z _ { 1 } - 5 z _ { 2 } & = 0 \\ 20 z _ { 1 } & = 5 z _ { 2 } \\ \dfrac { z _ { 1 } } { 1 } & = \dfrac { z _ { 2 } } { 4 } \end{aligned} \] Therefore, the set of
vectors that are orthogonal to $y$ have the form:
\[ \dfrac { z _ { 1 } } { 1 } = \dfrac { z _ { 2 } } { 4 } \]
Linear Algebra and its Applications, 6th Edition , By David C. Lay , Test Bank & Solutions Manual To get more information about this please send us E-mail to smtb5000 @gmail .co…
Test Bank & solutions manual
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