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Exercise 4 Determine the dimension of each of the following subspaces:
a. $U_{7}=\left\{\left(\begin{array}{l}{x} \\ {y}\end{array}\right) | x-2 y=0\right\}$

$x-2 y=0$

Let $y=t \Rightarrow x=2 t$

$\left(\begin{array}{l}{x} \\ {y}\end{array}\right) \Rightarrow\left(\begin{array}{c}{2 t} \\ {t}\end{array}\right) \Rightarrow\left(\begin{array}{l}{2} \\ {1}\end{array}\right) t$

$U_{7}=Span\left(\begin{array}{l}{2} \\ {1}\end{array}\right)$

$dim\left(U_{7}\right)=1$

b. $U_{8}=\left\{\left(\begin{array}{l}{x} \\ {y} \\ {z}\end{array}\right) | x-2 y-z=0\right\}$

$x-2 y-z=0$

Let $y=r$
Let $z=s$

$x=2 r+s$

$\left(\begin{array}{l}{x} \\ {y} \\ {z}\end{array}\right) \Rightarrow\left(\begin{array}{c}{2 r+s} \\ {r} \\ {s}\end{array}\right)=\left(\begin{array}{l}{2} \\ {1} \\ {0}\end{array}\right) r+\left(\begin{array}{l}{1} \\ {0} \\ {1}\end{array}\right) s$

$U_{8}=span\left(\left(\begin{array}{l}{2} \\ {1} \\ {0}\end{array}\right),\left(\begin{array}{l}{1} \\ {0} \\ {1}\end{array}\right)\right)$

$dim(U_{8})=2$

c. $U_{9}=\left\{\left(\begin{array}{l}{x} \\ {y} \\ {z}\end{array}\right) | x-2 y=0\right\}$

$x-2 y=0$

Let $y=r$
Let $z=s$

$x=2 r$

$\left(\begin{array}{l}{x} \\ {y} \\ {z}\end{array}\right) \Rightarrow\left(\begin{array}{l}{2 r} \\ {r} \\ {s}\end{array}\right)=\left(\begin{array}{l}{2} \\ {1} \\ {0}\end{array}\right) r+\left(\begin{array}{l}{0} \\ {0} \\ {1}\end{array}\right) s$

$U_{9}=span\left(\left(\begin{array}{l}{2} \\ {1} \\ {0}\end{array}\right),\left(\begin{array}{l}{0} \\ {0} \\ {1}\end{array}\right)\right)$

$\Rightarrow dim\left(U_{1}\right)=2$