Exercise4

Consider the following vectors in \(\real^3\text{:}\)

\begin{equation*} \vvec_1=\threevec{1}{3}{2}, \vvec_2=\threevec{0}{1}{4}, \vvec_3=\threevec{-2}{-5}{0}, \vvec_4=\threevec{-2}{-1}{-1}, \vvec_5=\threevec{1}{-2}{-1} \text{.} \end{equation*}
  1. Do these vectors form a basis for \(\real^3\text{?}\) Explain your thinking.

  2. Find a subset of these vectors that forms a basis of \(\real^3\text{.}\)

  3. Suppose you have a set of vectors \(\vvec_1, \vvec_2,\ldots,\vvec_6\) in \(\real^4\) such

    \begin{equation*} \left[\begin{array}{rrrr} \vvec_1 \amp \vvec_2 \amp \ldots \amp \vvec_6 \end{array}\right] \sim \left[\begin{array}{rrrrrr} 1 \amp 0 \amp -2 \amp 0 \amp 1 \amp 0 \\ 0 \amp 1 \amp 3 \amp 0 \amp -4 \amp 0 \\ 0 \amp 0 \amp 0 \amp 1 \amp 2 \amp 0 \\ 0 \amp 0 \amp 0 \amp 0 \amp 0 \amp 1 \\ \end{array}\right] \text{.} \end{equation*}

    Find a subset of the vectors that span \(\real^4\text{.}\)

in-context