Exercise 5

Suppose we have a set of vectors \(\vvec_1,\vvec_2,\vvec_3,\vvec_4\) in \(\real^5\) that satisfy the relationship:

\begin{equation*} 2\vvec_1 - \vvec_2 + 3\vvec_3 + \vvec_4 = \zerovec \end{equation*}

and suppose that \(A\) is the matrix \(A=\left[\begin{array}{rrrr} \vvec_1\amp\vvec_2\amp\vvec_3\amp\vvec_4 \end{array}\right] \text{.}\)

  1. Find a nontrivial solution to the equation \(A\xvec = \zerovec\text{.}\)

  2. Explain why the matrix \(A\) has a column without a pivot position.

  3. Write one of the vectors as a linear combination of the others.

  4. Explain why the set of vectors is linearly dependent.