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Determine whether the set $S$ is linearly independent or linearly dependent.
$$(a) S=\{(0,1,1),(1,0,1),(-1,0,0)\}$$

Let $$C_{1} V_{1}+C_{2} V_{2}+C_{3} V_{3}=0$$

$$C_{1}\left(\begin{array}{l}{0} \\ {1} \\ {1}\end{array}\right)+C_{2}\left(\begin{array}{l}{1} \\ {0} \\ {1}\end{array}\right)+C_{3}\left(\begin{array}{l}{1} \\ {0} \\ {0}\end{array}\right)=\left(\begin{array}{l}{0} \\ {0} \\ {0}\end{array}\right)$$

$$C_{2}-C_{3}=0$$
$$C_{1} \quad=0$$
$$C_{1}+C_{2}=0$$

$$\Rightarrow 0+c_{2}=0 \Rightarrow c_{2}=0$$

$$\Rightarrow 0-c_{3}=0 \Rightarrow c_{3}=0$$

The sct s is Linearly independent

$$(b) S=\{(-2,3,7),(4,-1,5),(3,1,3),(5,0,2)\} $$

Let $$c_{1} v_{1}+c_{2} v_{2}+c_{3} v_{3}+c_{4} v_{4}=0$$

$$\left(\begin{array}{cccc|c}{-2} & {4} & {3} & {5} & {0} \\ {3} & {-1} & {1} & {0} & {0} \\ {7} & {5} & {3} & {2} & {0}\end{array}\right) \Rightarrow\left(\begin{array}{cccc|c}{1} & {0} & {0} & {-13 / 37} & {0} \\ {0} & {1} & {0} & {6 / 37} & {0} \\ {0} & {0} & {1} & {45 / 37} & {0}\end{array}\right)$$

$$\Rightarrow$$ Inf, No, of sol. S is linearly dep.

Let $$C_{4}=t \Rightarrow C_{1}=13 / 37 t$$

$$C_{2}=-6 / 37 t$$
$$C_{3}=-45 / 37 t$$

(c) Let $$\left.v_{1}=(-2,3,7) \quad v_{2}=(4,-1,5) \quad v_{3}=(3,1,3) \quad v_{4}=(5,0,2)\right\}$$

i. Are $$v_{1} ; v_{2} ; v_{3}$$ linearly independent.
ii. Is $$v_{4} \in span\left\{v_{1}, v_{2}, v_{3}\right\}$$

$$\left(\begin{array}{cccc|c}{-2} & {4} & {3} & {5} & {0} \\ {3} & {-1} & {1} & {0} & {0} \\ {7} & {5} & {3} & {2} & {0}\end{array}\right) \Rightarrow\left(\begin{array}{cccc|c}{1} & {0} & {0} & {-13 / 37} & {0} \\ {0} & {1} & {0} & {6 / 37} & {0} \\ {0} & {0} & {1} & {45 / 37} & {0}\end{array}\right)$$

Let $$C_{4}=t \Rightarrow C_{1}=13 / 37 t$$

$$C_{2}=-6 / 37 t$$
$$C_{3}=-45 / 37 t$$

$$13 / 37{t}\left(\begin{array}{c}{-2} \\ {3} \\ {7}\end{array}\right)-6 / 37 t\left(\begin{array}{c}{4} \\ {-1} \\ {5}\end{array}\right)-45 / 37{t}\left(\begin{array}{c}{3} \\ {1} \\ {3}\end{array}\right)+t\left(\begin{array}{l}{5} \\ {0} \\ {2}\end{array}\right)=\left(\begin{array}{l}{0} \\ {0} \\ {0}\end{array}\right)$$

at $$t=1$$

$$\left(\begin{array}{l}{5} \\ {0} \\ {2}\end{array}\right)=-13 / 37\left(\begin{array}{c}{-2} \\ {3} \\ {7}\end{array}\right)+6 / 37\left(\begin{array}{c}{4} \\ {-1} \\ {5}\end{array}\right)+45 / 37\left(\begin{array}{c}{1} \\ {3} \\ {3}\end{array}\right)$$

$$\Rightarrow V_{4} \in span\left\{v_{1}, v_{2}, v_{3}\right\} $$

If the vectors $$u, v$$ and $$w$$ are linearly independent, will the vectors $$u+v, v+w$$ and
$$u+w$$ also be linearly independent? Justify your answer.

$$a(u+v)+b(v+w)+c(u+w)=0 \Rightarrow a, b, c=0$$

$$a u+a v+b v+b w+c u+c w=0$$
$$(a+c) u+(a+b) v+(b+c) w=0$$

∵ $$u, v, w$$ are $$L \cdot I$$

$$(a+c)=0 \Rightarrow c = -a \Rightarrow c=0$$
$$(a+b)=0 \Rightarrow b = -a \Rightarrow b=0$$ 
$$(b+c)=0 \Rightarrow-a-a=0 \Rightarrow-2 a=0 \Rightarrow a=0$$

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