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

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

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

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

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

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