Question

Short answer

The vectors are linearly dependent for all the possible values of \(h\).

Step-by-step solution

Set of two or more vectors

When a set has more vectors than entries in each vector, it is said to be linearly dependent.

Let \({v_1},{v_2}\,\),and \({v_3}\) be the three vectors. The linear dependence of these three vectors in the form of an augmented matrix is \(\left[ {\begin{array}{*{20}{c}}{{v_1}}&{{v_2}}&{{v_3}}&0\end{array}} \right]\).

Hence, the augmented matrix is:

\(\left[ {\begin{array}{*{20}{c}}2&{ - 6}&8&0\\{ - 4}&7&h&0\\1&{ - 3}&4&0\end{array}} \right]\)

Reduce the matrix into an echelon

Apply row operation \({R_2} \to {R_2} + 2{R_1}\) to the augmented matrix above.

\(\left[ {\begin{array}{*{20}{c}}2&{ - 6}&8&0\\0&{ - 5}&{h + 16}&0\\1&{ - 3}&4&0\end{array}} \right]\)

Apply row operation \({R_3} \to {R_3} - \frac{1}{2}{R_1}\).

\(\left[ {\begin{array}{*{20}{c}}2&{ - 6}&8&0\\0&{ - 5}&{h + 16}&0\\0&0&0&0\end{array}} \right]\)

Echelon matrix

The pivots in the echelon matrix are represented as:

Linear dependent equation

If the zero vector appears in a set \(S = \left\{ {{v_1},{v_2},.....{v_p}} \right\}\) in \({R^n}\) , the set is linearly dependent.

Thus, the equation can be written as \({x_1}{v_1} + {x_2}{v_2} + {x_3}{v_3} = 0\). The vector has a free variable, and does not depend upon the value of \(h\).

Hence, the vectors are linearly dependent for all the possible values of \(h\).