Linear Independence

1. Determine if the vectors $\vec{v_1} = \left[\begin{array}{r} 0 \\ 0 \\ 2 \end{array}\right], \vec{v_2} = \left[\begin{array}{r} 0 \\ 5 \\ 1 \end{array}\right], \vec{v_3} = \left[\begin{array}{r} 2 \\ 1 \\ -8 \end{array}\right]$ are linearly independent.   solution

2. Find the value(s) of $h$ for which the vectors are linearly dependent.

(a) $\left[\begin{array}{r} -2 \\ 7 \\ 1 \end{array}\right], \left[\begin{array}{r} 8\\ 5 \\ -4 \end{array}\right], \left[\begin{array}{r} 6 \\ h \\ -3 \end{array}\right]$    solution     (b) $\left[\begin{array}{r} 1\\ -1 \\ 3 \end{array}\right], \left[\begin{array}{r} -5\\ 7 \\ 8 \end{array}\right], \left[\begin{array}{r} 1 \\ 1 \\ h \end{array}\right]$    solution

3. Determine by inspection whether the vectors are linearly independent. Justify your answer.  solution

(a) $\left[\begin{array}{r} {3} \\ {-1} \end{array}\right],\left[\begin{array}{r} {2} \\ {8} \end{array}\right],\left[\begin{array}{r} {1} \\ {3} \end{array}\right]$    (b) $\left[\begin{array}{r} {2} \\ {5} \\ {-1} \end{array}\right],\left[\begin{array}{r} {0} \\ {0} \\ {0} \end{array}\right],\left[\begin{array}{r} {-6} \\ {5} \\ {1} \end{array}\right]$    (c) $\left[\begin{array}{r} {1} \\ {6} \\ {-3} \end{array}\right],\left[\begin{array}{r} {-5} \\ {2} \\ {1} \end{array}\right]$    (d) $\left[\begin{array}{r} {-12} \\ {6} \\ {3} \end{array}\right],\left[\begin{array}{r} {-8} \\ {4} \\ {2} \end{array}\right]$

4. Suppose that $S=\{ \overrightarrow{v_1}, \overrightarrow{v_2}, \overrightarrow{v_3}\}$ is a linearly independent set of vectors in a vector space. Is $W=\{ \overrightarrow{w_1}, \overrightarrow{w_2}, \overrightarrow{w_3}\}$, where $\overrightarrow{w_1}=\overrightarrow{v_1}-\overrightarrow{v_2}, \quad \overrightarrow{w_2}=\overrightarrow{v_1}-\overrightarrow{v_3}, \quad \overrightarrow{w_3}=\overrightarrow{v_2}-\overrightarrow{v_3}$,   linearly dependent or linearly independent? Justify your answer.   solution