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 16E from Chapter 1.3 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 16E
Chapter:
Problem:
List five vectors in Span {v1, v2}. For each vector, show the weights on v1 and v2 used to generate the vector and list the three entries of the vector. Do not make a sketch.
Step-by-Step Solution
Given Information
We are given that two vectors\[{v_1} = \left[ {\begin{array}{*{20}{l}}3\\0\\2\end{array}} \right],{v_2} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\0\\3\end{array}} \right]\]We have to list five vectors in the span{$v_1,v_2$}
Step-1: Five Vectors
Let us choose five sets of weights as:\[w = \left\{ {0,0} \right\},\left\{ {1,0} \right\},\left\{ {0,1} \right\},\left\{ {1,1} \right\},\left\{ {1, - 1} \right\}\]So, the five vectors are:\[\begin{array}{l}{V_1} = 0{v_1} + 0{v_2} = \left[ {\begin{array}{*{20}{c}}0\\0\\0\end{array}} \right]\\{V_2} = 1{v_1} + 0{v_2} = \left[ {\begin{array}{*{20}{l}}3\\0\\2\end{array}} \right]\\{V_3} = 0{v_1} + 1{v_2} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\0\\3\end{array}} \right]\\{V_4} = 1{v_1} + 1{v_2} = \left[ {\begin{array}{*{20}{c}}{3 - 2}\\0\\{2 + 3}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\0\\5\end{array}} \right]\\{V_5} = 1{v_1} - 1{v_2} = \left[ {\begin{array}{*{20}{c}}{3 + 2}\\0\\{2 - 3}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}5\\0\\{ - 1}\end{array}} \right]\end{array}\]Therefore,
\[\left\{ {\left[ {\begin{array}{*{20}{c}}0\\0\\0\end{array}} \right],\left[ {\begin{array}{*{20}{l}}3\\0\\2\end{array}} \right],\left[ {\begin{array}{*{20}{c}}{ - 2}\\0\\3\end{array}} \right],\left[ {\begin{array}{*{20}{c}}1\\0\\5\end{array}} \right],\left[ {\begin{array}{*{20}{c}}5\\0\\{ - 1}\end{array}} \right]} \right\}\]
We are given that two vectors\[{v_1} = \left[ {\begin{array}{*{20}{l}}3\\0\\2\end{array}} \right],{v_2} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\0\\3\end{array}} \right]\]We have to list five vectors in the span{$v_1,v_2$}
Step-1: Five Vectors
Let us choose five sets of weights as:\[w = \left\{ {0,0} \right\},\left\{ {1,0} \right\},\left\{ {0,1} \right\},\left\{ {1,1} \right\},\left\{ {1, - 1} \right\}\]So, the five vectors are:\[\begin{array}{l}{V_1} = 0{v_1} + 0{v_2} = \left[ {\begin{array}{*{20}{c}}0\\0\\0\end{array}} \right]\\{V_2} = 1{v_1} + 0{v_2} = \left[ {\begin{array}{*{20}{l}}3\\0\\2\end{array}} \right]\\{V_3} = 0{v_1} + 1{v_2} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\0\\3\end{array}} \right]\\{V_4} = 1{v_1} + 1{v_2} = \left[ {\begin{array}{*{20}{c}}{3 - 2}\\0\\{2 + 3}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\0\\5\end{array}} \right]\\{V_5} = 1{v_1} - 1{v_2} = \left[ {\begin{array}{*{20}{c}}{3 + 2}\\0\\{2 - 3}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}5\\0\\{ - 1}\end{array}} \right]\end{array}\]Therefore,
\[\left\{ {\left[ {\begin{array}{*{20}{c}}0\\0\\0\end{array}} \right],\left[ {\begin{array}{*{20}{l}}3\\0\\2\end{array}} \right],\left[ {\begin{array}{*{20}{c}}{ - 2}\\0\\3\end{array}} \right],\left[ {\begin{array}{*{20}{c}}1\\0\\5\end{array}} \right],\left[ {\begin{array}{*{20}{c}}5\\0\\{ - 1}\end{array}} \right]} \right\}\]