Question

In Exercises 10 through 20 , use paper and pencil to identify the redundant vectors. Thus determine whether the given vectors are linearly independent.

14.$\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]\mathbf{,}\left[\begin{array}{c}3\\ 2\\ 1\end{array}\right]\mathbf{,}\left[\begin{array}{c}6\\ 5\\ 4\end{array}\right]$ .

Short answer

The vectors $\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right],\left[\begin{array}{c}3\\ 2\\ 1\end{array}\right],\left[\begin{array}{c}6\\ 5\\ 4\end{array}\right]$are redundant and linearly dependent.

Step-by-step solution

Consider the set.

The vectors ${\overrightarrow{v}}_1,{\overrightarrow{v}}_2,...,{\overrightarrow{v}}_m$in ${\mathrm{ℝ}}^n$are linearly dependent if and only if there are nontrivial relation among them.

If linearly dependent, then, the vectors are said to be redundant.

Consider the vectors,$\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right],\left[\begin{array}{c}3\\ 2\\ 1\end{array}\right],\left[\begin{array}{c}6\\ 5\\ 4\end{array}\right]$

Check for linear independency of vectors.

The reduced form of the matrix is,

$$\begin{gathered} \left(\begin{array}{ccc}1& 3& 6\\ 1& 2& 5\\ 1& 1& 4\end{array}\right) \\ =\left(\begin{array}{ccc}1& 3& 6\\ 0& -2& -2\\ 0& 0& 0\end{array}\right) \\ =\left(\begin{array}{ccc}1& 0& 3\\ 0& 1& 1\\ 0& 0& 0\end{array}\right) \end{gathered}$$

Solve the above equations.

$v_1=\left[\left.\begin{array}{c}1\\ 0\\ 0\end{array}\right],v_2=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right],v_3=\left[\begin{array}{c}3\\ 1\\ 0\end{array}\right.\right]$

role="math" localid="1660738737357" $V_3=3v_1+v_2$

Final answer.

The vectors $\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right],\left[\begin{array}{c}3\\ 2\\ 1\end{array}\right],\left[\begin{array}{c}6\\ 5\\ 4\end{array}\right]$ are redundant and linearly dependent.