Linear Algebra and Its Applications (5th Edition)

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

Chapter 1 - Linear Equations in Linear Algebra - Problems: 14

Answer

$b$ is a linear combination of the columns of the matrix $A$.

Work Step by Step

The question posed is equivalent to asking whether the system represented by the augmented matrix below is consistent: $$ \begin{bmatrix} 1 & -2 & -6 & 11 \\ 0 & 3 & 7 & -5 \\ 1 & -2 & 5 & 9 \end{bmatrix} $$ We can determine this via row reduction. Subtract the first row from the third: $$ \begin{bmatrix} 1 & -2 & -6 & 11 \\ 0 & 3 & 7 & -5 \\ 0 & 0 & 11 & -2 \end{bmatrix} $$ We could continue the row reduction process and find the reduced echelon form of the matrix, but at this point it is clear that there are pivot entries in each row, and thus there will be no row that reduces to $0=1$. Hence, the system is consistent and $b$ is a linear combination of the columns of the matrix $A$.
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