Question

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

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

Short answer

The vectors $\left[\begin{array}{c}2\\ 1\end{array}\right],\left[\begin{array}{c}6\\ 3\end{array}\right]$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,$\left[\begin{array}{c}2\\ 1\end{array}\right],\left[\begin{array}{c}6\\ 3\end{array}\right]$

Check for linear independency of vectors.

The reduced form of the matrix is,

$$\begin{gathered} \left(\begin{array}{cc}2& 6\\ 1& 3\end{array}\right) \\ =\left(\begin{array}{cc}2& 6\\ 0& 0\end{array}\right) \\ =\left(\begin{array}{cc}1& 3\\ 0& 0\end{array}\right) \end{gathered}$$

Solve the above equations.

$V_1=\left[\left.\begin{array}{c}1\\ 0\end{array}\right],V_2=\left[\begin{array}{c}3\\ 0\end{array}\right.\right]$

$V_2=3v_1$

Final answer.

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