Exercise 1

Consider the set of vectors

\begin{equation*} \vvec_1 = \threevec{1}{2}{1}, \vvec_2 = \threevec{0}{1}{3}, \vvec_3 = \threevec{2}{3}{-1}, \vvec_4 = \threevec{-2}{4}{-1}\text{.} \end{equation*}
  1. Explain why this set of vectors is linearly dependent.

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

  3. Find weights \(c_1\text{,}\) \(c_2\text{,}\) \(c_3\text{,}\) and \(c_4\text{,}\) not all of which are zero, such that

    \begin{equation*} c_1\vvec_1 + c_2 \vvec_2 + c_3 \vvec_3 + c_4 \vvec_4 = \zerovec\text{.} \end{equation*}
  4. Find a nontrivial solution to the homogenous equation \(A\xvec = \zerovec\) where \(A=\left[\begin{array}{rrrr} \vvec_1\amp\vvec_2\amp\vvec_3\amp\vvec_4 \end{array}\right]\text{.}\)