Math  /  Algebra

QuestionFor each of the following sets of vectors, determine whether it is linearly independent or linearly dependent. If it is dependent, give a non-trivial linear combination of the vectors yielding the zero vector. Give your combination as an expression using u,vu, v, and ww for the vector variables u,v\mathbf{u}, \mathbf{v}, and w\mathbf{w}. a) u=[322]v=[967]w=[344]\mathbf{u}=\left[\begin{array}{c}3 \\ 2 \\ -2\end{array}\right] \mathbf{v}=\left[\begin{array}{c}-9 \\ -6 \\ 7\end{array}\right] \quad \mathbf{w}=\left[\begin{array}{c}3 \\ 4 \\ -4\end{array}\right] < Select an answer > b) u=[010]v=[1011]w=[1011]\mathbf{u}=\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right] \mathbf{v}=\left[\begin{array}{c}10 \\ 1 \\ -1\end{array}\right] \mathbf{w}=\left[\begin{array}{c}-10 \\ -1 \\ 1\end{array}\right] < Select an answer >

Studdy Solution
Solve the system of equations. From the first equation:
10c210c3=0 10c_2 - 10c_3 = 0 c2=c3 c_2 = c_3
Substitute c2=c3 c_2 = c_3 into the second equation:
c1+c2c3=0 c_1 + c_2 - c_3 = 0 c1=0 c_1 = 0
Substitute c2=c3 c_2 = c_3 into the third equation:
c2+c3=0 -c_2 + c_3 = 0 c2=c3 c_2 = c_3
Since c2=c3 c_2 = c_3 and c1=0 c_1 = 0 , we have a non-trivial solution by choosing c2=1 c_2 = 1 and c3=1 c_3 = 1 .
The vectors are linearly dependent, and a non-trivial linear combination is:
0u+1v+1w=0 0u + 1v + 1w = 0
The set (a) is linearly independent. The set (b) is linearly dependent with the combination v+w=0 v + w = 0 .

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