Elementary Linear Algebra 7th Edition

Published by Cengage Learning
ISBN 10: 1-13311-087-8
ISBN 13: 978-1-13311-087-3

Chapter 4 - Vector Spaces - 4.1 Vectors in Rn - 4.1 Exercises - Page 153: 48

Answer

$v$ can not be written as a linear combination of the vectors $u_1,u_2,u_3$.

Work Step by Step

Suppose the linear combination $v= au_1+bu_2+cu_3$, then we have $$(0,5,-4,0)=a(1,3,2,1)+b(2,-2,-5,4)+c(2,-1,3,6), \quad a,b,c\in R.$$ Which yields the following system of equations \begin{align*} a+2b+2c&=0\\ 3a-2b-c&=5\\ 2a-5b+3c&=-4\\ a+4b+6c&=0. \end{align*} The augmented matrix is given by $$\left[ \begin {array}{cccc} 1&2&2&0\\ 3&-2&-1&5 \\ 2&-5&3&-4\\ 1&4&6&0\end {array} \right] $$ Using Gauss-eleminition, we get $$\left[ \begin {array}{cccc} 1&2&2&0\\ 0&-8&-7&5 \\ 0&0&{\frac {55}{8}}&-{\frac {77}{8}} \\ 0&0&0&{\frac {22}{5}}\end {array} \right] $$ which shows that the system has no solution. , So $v$ can not be written as a linear combination of the vectors $u_1,u_2,u_3$.
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