Linear Algebra and Its Applications (5th Edition)

Published by Pearson
ISBN 10: 032198238X
ISBN 13: 978-0-32198-238-4

Chapter 1 - Linear Equations in Linear Algebra - 1.9 Exercises - Page 79: 17

Answer

See solution

Work Step by Step

\[ T(\mathbf{x})=\left[\begin{array}{c} 0 \\ x_{1}+x_{2} \\ x_{2}+x_{3} \\ x_{3}+x_{4} \end{array}\right]=A\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \] To express $T(x)$ as $A x,$ write $T(x)$ and $x$ as column vectors, and then fill in the entries in $A$ by inspection \[ T(\mathbf{x})=\left[\begin{array}{llll} 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \] Observe that since $T(x)$ and $x$ have four entries, $A$ must be a $4 \times 4$ matrix
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