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