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 27E from Chapter 1.4 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 27E
Chapter:
Problem:
Let q1, q2, q3, and v represent vectors in ℝ5, and let x1, x2, and x3 denote scalars. Write the following vector equation as a matrix equation. Identify any symbols you choose to use.
Step-by-Step Solution
Given Information
We are given that $\mathbf { q } _ { 1 } , \mathbf { q } _ { 2 } , \mathbf { q } _ { 3 } , \text { and } \mathbf { v }$ are in $R^5$ and $x_1, x_2$ and $x_3$ are scalars. We have to write the vector equation $x _ { 1 } \mathbf { a } _ { 1 } + \cdots + x _ { n } \mathbf { a } _ { n } = \mathbf { b }$ in matrix form.
Step-1:
The vector equation $x _ { 1 } a _ { 1 } + \dots + x _ { n } a _ { n } = b$ can be written as: \[ \left[ \begin{array} { l l l l } { \mathbf { a } _ { 1 } } & { \mathbf { a } _ { 2 } } & { \cdots } & { \mathbf { a } _ { n } } \end{array} \right] \left[ \begin{array} { c } { x _ { 1 } } \\ { \vdots } \\ { x _ { n } } \end{array} \right] = \mathbf { b } \] So, the vector equation can be written as:
\[\left[ {\begin{array}{*{20}{c}} {{{\bf{q}}_1}}&{{{\bf{q}}_2}}&{{{\bf{q}}_3}} \end{array}} \right]\left[ {\begin{array}{*{20}{l}} {{x_1}}\\ {{x_2}}\\ {{x_3}} \end{array}} \right] = {\bf{v}} \Rightarrow \,Q{\bf{x}} = {\bf{v}}\]
We are given that $\mathbf { q } _ { 1 } , \mathbf { q } _ { 2 } , \mathbf { q } _ { 3 } , \text { and } \mathbf { v }$ are in $R^5$ and $x_1, x_2$ and $x_3$ are scalars. We have to write the vector equation $x _ { 1 } \mathbf { a } _ { 1 } + \cdots + x _ { n } \mathbf { a } _ { n } = \mathbf { b }$ in matrix form.
Step-1:
The vector equation $x _ { 1 } a _ { 1 } + \dots + x _ { n } a _ { n } = b$ can be written as: \[ \left[ \begin{array} { l l l l } { \mathbf { a } _ { 1 } } & { \mathbf { a } _ { 2 } } & { \cdots } & { \mathbf { a } _ { n } } \end{array} \right] \left[ \begin{array} { c } { x _ { 1 } } \\ { \vdots } \\ { x _ { n } } \end{array} \right] = \mathbf { b } \] So, the vector equation can be written as:
\[\left[ {\begin{array}{*{20}{c}} {{{\bf{q}}_1}}&{{{\bf{q}}_2}}&{{{\bf{q}}_3}} \end{array}} \right]\left[ {\begin{array}{*{20}{l}} {{x_1}}\\ {{x_2}}\\ {{x_3}} \end{array}} \right] = {\bf{v}} \Rightarrow \,Q{\bf{x}} = {\bf{v}}\]