Determine if the vectors
| \(\Vect{a}\) | \(=\) | \((1,2,3,0)\) |
| \(\Vect{b}\) | \(=\) | \((1,1,2,2)\) |
| \(\Vect{c}\) | \(=\) | \((3,1,4,1)\) |
| \(\Vect{d}\) | \(=\) | \((2,1,3,0)\) |
of \(\RNrSpc{4}\) are linearly independent.
We form the matrix
\[ \Mtrx{A} = \left[ \begin{array}{cccc} 1 & 1 & 3 & 2 \\ 2 & 1 & 1 & 1 \\ 3 & 2 & 4 & 3 \\ 0 & 2 & 1 & 0 \end{array} \right] \]which has the given vectors as its column vectors. The RREF of \(\Mtrx{A}\) is
\[ \begin{array}{rrrr} 1 & 0 & 0 & 1/3 \\ 0 & 1 & 0 & -1/3 \\ 0 & 0 & 1 & 2/3 \\ 0 & 0 & 0 & 0 \end{array} \]So \( \Rnk{\Mtrx{A}} = 3 < 4 \). Therefore, the vectors are linearly dependent.
Determine if the vectors
| \(\Vect{w}\) | \(=\) | \((3,1,1,-1,2,6,1,1)\) |
| \(\Vect{x}\) | \(=\) | \((0,4,9,6,2,6,3,-2)\) |
| \(\Vect{y}\) | \(=\) | \((2,0,0,2,3,1,-1,0)\) |
| \(\Vect{z}\) | \(=\) | \((1,1,-2,-1,3,1,1,4)\) |
of \(\RNrSpc{8}\) are linearly independent.
If we use the given vectors as the column vectors of a matrix and determine its RREF, we find
\[ \Mtrx{A} = \left[ \begin{array}{rrrr} 3 & 0 & 2 & 1 \\ 1 & 4 & 0 & 1 \\ 1 & 9 & 0 & -2 \\ -1 & 6 & 2 & -1 \\ 2 & 2 & 3 & 3 \\ 6 & 6 & 1 & 1 \\ 1 & 3 & -1 & 1 \\ 1 & -2 & 0 & 4 \end{array} \right]\qquad \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \]Thus \(\Rnk{\Mtrx{A}} = 4\), implying that its columns are linearly independent. So \(\Vect{w},\Vect{x},\Vect{y},\Vect{z}\) are linearly independent.