Question 4.E.3.1: Determine which of the following sets are linearly independe......
Determine which of the following sets are linearly independent. For those sets that are linearly dependent, write one of the vectors as a linear combination of the others.
(a) \begin{Bmatrix}\begin{pmatrix}1\\2\\3\end{pmatrix}, \begin{pmatrix}2\\1\\0\end{pmatrix}, \begin{pmatrix}1\\5\\9\end{pmatrix} \end{Bmatrix},
(b) {( 1 2 3), ( 0 4 5), ( 0 0 6), ( 1 1 1)},
(c) \begin{Bmatrix}\begin{pmatrix}3\\2\\1\end{pmatrix}, \begin{pmatrix}1\\0\\0\end{pmatrix}, \begin{pmatrix}2\\1\\0 \end{pmatrix} \end{Bmatrix},
(d) {( 2 2 2 2), ( 2 2 0 2), ( 2 0 2 2)} ,
(e) \begin{Bmatrix}\begin{pmatrix}1\\2\\0\\4\\0\\3\\0\end{pmatrix}, \begin{pmatrix}0\\2\\0\\4\\1\\3\\0\end{pmatrix}, \begin{pmatrix}0\\2\\1\\4\\0\\3\\0\end{pmatrix}, \begin{pmatrix}0\\2\\0\\4\\0\\3\\1\end{pmatrix}\end{Bmatrix}.
(a) and (b) are linearly dependent—all others are linearly independent. To write one vector as a combination of others in a dependent set, place the vectors as columns in A and find E_{A}. This reveals the dependence relationships among columns of A.