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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 $$u_{2}$$

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