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• Notes

Exercise $1 :$ Which of the following sets of vectors forms a basis of $R^{2}$

a. $\left\{\left(\begin{array}{l}{1} \\ {0}\end{array}\right)\right\}$ X

b. $\left\{\left(\begin{array}{l}{1} \\ {0}\end{array}\right),\left(\begin{array}{l}{1} \\ {2}\end{array}\right),\left(\begin{array}{l}{0} \\ {1}\end{array}\right)\right\}$ X

c. $\left\{\left(\begin{array}{l}{2} \\ {4}\end{array}\right),\left(\begin{array}{l}{1} \\ {2}\end{array}\right)\right\}$

$\left(\begin{array}{l}{2} \\ {4}\end{array}\right)=2\left(\begin{array}{l}{1} \\ {2}\end{array}\right)$ X

d. $\left\{\left(\begin{array}{c}{-2} \\ {3}\end{array}\right),\left(\begin{array}{l}{1} \\ {2}\end{array}\right)\right\}$

$\left(\begin{array}{cc|c}{1} & {-2} & {0} \\ {2} & {3} & {0}\end{array}\right)$

Exercise $2 :$ Which of the following sets of vectors forms a basis of $R^{3} .$

a. $\left\{\left(\begin{array}{c}{-1} \\ {2} \\ {0}\end{array}\right)\right\}$ X

b. $\left\{\left(\begin{array}{c}{-1} \\ {2} \\ {0}\end{array}\right),\left(\begin{array}{l}{0} \\ {0} \\ {0}\end{array}\right),\left(\begin{array}{l}{1} \\ {2} \\ {3}\end{array}\right)\right\}$ X

c. $\left\{\left(\begin{array}{c}{\sqrt{-1}} \\ {0} \\ {0}\end{array}\right),\left(\begin{array}{c}{0} \\ {\sqrt{-1}} \\ {0}\end{array}\right),\left(\begin{array}{c}{0} \\ {0} \\ {\sqrt{-1}}\end{array}\right)\right\}$

$\sqrt{-1} \notin R$

$\left(\begin{array}{c}{\sqrt{-1}} \\ {0} \\ {0} \end{array}\right) \notin R^{3}$

Exercise 3 Determine the dimension of each of the following subspaces:

a. $U_{3}=\text {span}\left(\left(\begin{array}{c}{1} \\ {3} \\ {-1}\end{array}\right),\left(\begin{array}{l}{2} \\ {1} \\ {0}\end{array}\right)\right)$

$\Rightarrow$ The set of vectors are linearly Independent

$\text {dim}\left(U_{3}\right)=2$

$\left(\begin{array}{l}{2} \\ {1} \\ {0}\end{array}\right) \neq n\left(\begin{array}{c}{1} \\ {3} \\ {-1}\end{array}\right)$

$\left(\begin{array}{r}{1} \\ {3} \\ {-1}\end{array}\right) \neq n\left(\begin{array}{l}{2} \\ {1} \\ {0}\end{array}\right)$

b. $U_{4}=\text{span}\left(\left(\begin{array}{c}{0} \\ {-2} \\ {1}\end{array}\right),\left(\begin{array}{c}{1} \\ {4} \\ {-1}\end{array}\right),\left(\begin{array}{c}{-1} \\ {2} \\ {2}\end{array}\right)\right)$

$\left(\begin{array}{ccc|c}{0} & {1} & {-1} & {0} \\ {-2} & {4} & {2} & {0} \\ {1} & {-1} & {2} & {0}\end{array}\right)^{R_{1} \leftrightarrow R_{3}}$

$\left(\begin{array}{ccc|c}{1} & {-1} & {2} & {0} \\ {-2} & {4} & {2} & {0} \\ {0} & {1} & {-1} & {0}\end{array}\right) R_{2} \rightarrow R_{2}+2 R_{1}$

$\left(\begin{array}{ccc|c}{1} & {-1} & {2} & {0} \\ {0} & {2} & {6} & {0} \\ {0} & {1} & {-1} & {0}\end{array}\right) R_{2} \rightarrow{1 / 2} R_{2}$

$\left(\begin{array}{ccc|c}{1} & {-1} & {2} & {0} \\ {0} & {1} & {3} & {0} \\ {0} & {1} & {-1} & {0}\end{array}\right) R_{3} \rightarrow R_{3}-R_{2}$

$\left(\begin{array}{ccc|c}{1} & {-1} & {2} & {0} \\ {0} & {1} & {3} & {0} \\ {0} & {0} & {-4} & {0}\end{array}\right)$

$c_{1}, c_{2}, c_{3}=0$
3 vectors are L.I

$\text{dim}\left(U_{4}\right)=3$

c. $U_{5}=\text {span}\left(\left(\begin{array}{c}{-1} \\ {1} \\ {1}\end{array}\right),\left(\begin{array}{l}{2} \\ {0} \\ {1}\end{array}\right),\left(\begin{array}{c}{-4} \\ {2} \\ {1}\end{array}\right)\right)$

$\left(\begin{array}{ccc|c}{-1} & {2} & {-4} & {0} \\ {1} & {0} & {2} & {0} \\ {1} & {1} & {1} & {0}\end{array}\right) \begin{array}{l}{R_{2} \rightarrow R_{2}+R_{1}} \\ {R_{3} \rightarrow R_{3}+R_{1}}\end{array}$

$\left(\begin{array}{ccc|c}{-1} & {2} & {-4} & {0} \\ {0} & {2} & {-2} & {0} \\ {0} & {3} & {-3} & {0}\end{array}\right) \begin{array}{l} R_{1} \rightarrow -R_{1} \\ R_{2} \rightarrow 1 / 2 R_{2} \end{array}$

$\left(\begin{array}{ccc|c}{1} & {-2} & {4} & {0} \\ {0} & {1} & {-1} & {0} \\ {0} & {3} & {-3} & {0}\end{array}\right) R_{3} \rightarrow R_{3}-3R_{2}$

$\left(\begin{array}{ccc|c}{1} & {-2} & {4} & {0} \\ {0} & {1} & {-1} & {0} \\ {0} & {0} & {0} & {0}\end{array}\right)$

$dim (U_{5})=2$

d. $U_{6}=\text{span}\left(\left(\begin{array}{c}{1} \\ {1} \\ {1} \\ {1}\end{array}\right),\left(\begin{array}{l}{1} \\ {2} \\ {3} \\ {4}\end{array}\right),\left(\begin{array}{l}{4} \\ {3} \\ {2} \\ {1}\end{array}\right),\left(\begin{array}{l}{5} \\ {5} \\ {5} \\ {5}\end{array}\right)\right)$

$\left(\begin{array}{llll|l}{1} & {1} & {4} & {5} & {0} \\ {1} & {2} & {3} & {5} & {0} \\ {1} & {3} & {2} & {5} & {0} \\ {1} & {4} & {1} & {5} & {0}\end{array}\right) \begin{array}{l}{R_{2} \rightarrow R_{2}-R_{1}} \\ {R_{3} \rightarrow R_{3}-R_{1}} \\ {R_{4} \rightarrow R_{4}-R_{1}}\end{array}$

$\left(\begin{array}{cccc|c}{1} & {1} & {4} & {5} & {0} \\ {0} & {1} & {-1} & {0} & {0} \\ {0} & {2} & {-2} & {0} & {0} \\ {0} & {3} & {-3} & {0} & {0}\end{array}\right) \begin{array}{l} R_{3} \rightarrow R_{3}-2 R_{2} \\ R_{4} \rightarrow R_{4}-3R_{2} \end{array}$

$\left(\begin{array}{cccc|c}{1} & {1} & {4} & {5} & {0} \\ {0} & {1} & {-1} & {0} & {0} \\ {0} & {0} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0} & {0}\end{array}\right)$

$\text{dim}\left(U_{6}\right)=2$