Question

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

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

Short answer

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

Step-by-step solution

Consider the set.

The vectors ${\overrightarrow{v}}_1,{\overrightarrow{v}}_2,....,{\overrightarrow{v}}_m\mathrm{in}{\mathrm{ℝ}}^{\mathrm{n}}$in 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,$\left[\begin{array}{c}1\\ 2\end{array}\right],\left[\begin{array}{c}2\\ 3\end{array}\right],\left[\begin{array}{c}3\\ 4\end{array}\right]$

Check for linear independency of vectors.

The reduced form of the matrix is,

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

Solve the above equations.

$$\begin{gathered} v_1=\left[\begin{array}{c}1\\ 0\end{array}\right],v_2=\left[\begin{array}{c}0\\ 1\end{array}\right],v_3=\left[\begin{array}{c}-1\\ 2\end{array}\right] \\ {v}_3=-v_1+2v_2 \end{gathered}$$

Final answer.

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