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• Notes

In each of the following, verify whether the vector $y$ is a linear combination of the
other vectors:
a) $y=\left(\begin{array}{l}{5} \\ {1}\end{array}\right), u_{1}=\left(\begin{array}{c}{-1} \\ {1}\end{array}\right), u_{2}=\left(\begin{array}{l}{1} \\ {1}\end{array}\right)$

$y=a u_{1}+b u_{2}$

$\left(\begin{array}{l}{5} \\ {1}\end{array}\right)=a\left(\begin{array}{c}{-1} \\ {1}\end{array}\right)+b\left(\begin{array}{l}{1} \\ {1}\end{array}\right)$

$-a+b=5$
$a+b=1$
$2 b=6$
$b=3$

$\Rightarrow a+3=1$

$a=-2$

$-a+b=5 \Rightarrow-(-2)+3=5$

$y=-2\left(\begin{array}{c}{-1} \\ {1}\end{array}\right)+3\left(\begin{array}{l}{1} \\ {1}\end{array}\right)=\left(\begin{array}{l}{5} \\ {1}\end{array}\right)$

y is a linear combination of $u_{1}$ and $u_{2}$

$\left(\begin{array}{cc|c}{-1} & {1} & {5} \\ {1} & {1} & {1}\end{array}\right) R_{2} \rightarrow R_{2}+R_{1}$

$-a+b=5$
$2 b =6 \Rightarrow b=3$
$-a+3 =5 \Rightarrow a=-2$

In each of the following, verify whether the vector $y$ is a linear combination of the
other vectors:
b) $y=\left(\begin{array}{c}{5} \\ {3}\end{array}\right), u_{1}=\left(\begin{array}{c}{-1} \\ {-2}\end{array}\right), u_{2}=\left(\begin{array}{c}{1} \\ {2}\end{array}\right)$

$y=a u_{1}+b u_{2}$

$\left(\begin{array}{cc|c}{-1} & {1} & {5} \\ {-2} & {2} & {3}\end{array}\right)^{R_{1} \rightarrow -R_{2}}$

$\left(\begin{array}{cc|c}{1} & {-1} & {-5} \\ {-2} & {2} & {3}\end{array}\right) R_{2} \rightarrow R_{2}+2 R_{1}$

$\left(\begin{array}{cc|c}{1} & {-1} & {-5} \\ {0} & {0} & {-7}\end{array}\right) \Rightarrow 0=-7$ (false eq)

∵  The system has no sol.

$y$ can not be a linear combination from $u_{1}$ and $u_{2}$

In each of the following, verify whether the vector $y$ is a linear combination of the
other vectors:
C) $y=\left(\begin{array}{c}{5} \\ {10}\end{array}\right), u_{1}=\left(\begin{array}{c}{-1} \\ {-2}\end{array}\right), u_{2}=\left(\begin{array}{c}{1} \\ {2}\end{array}\right)$

$y=a u_{1}+b u_{2}$

$\left(\begin{array}{cc|c}{-1} & {1} & {5} \\ {-2} & {2} & {10}\end{array}\right)^{R_{1} \rightarrow -R_{1}}$

$\left(\begin{array}{cc|c}{1} & {-1} & {-5} \\ {-2} & {2} & {10}\end{array}\right) R_{2} \rightarrow R_{2}+2 R_{1}$

$\left(\begin{array}{cc|c}{1} & {-1} & {-5} \\ {0} & {0} & {0}\end{array}\right)$

Let $b=t$

$a=-5+t$

$\left(\begin{array}{l}{a} \\ {b}\end{array}\right)=\left(\begin{array}{c}{-5+t} \\ {t}\end{array}\right)$

∵  the system has Infinit No. of sol.

$y$ is a $L . C .$ of $u_{1}$ and $u_{2}$