Preview Activity3.5.1

Let's consider the following matrix \(A\) and its reduced row echelon form.

\begin{equation*} A = \left[\begin{array}{rrrr} 2 \amp -1 \amp 2 \amp 3 \\ 1 \amp 0 \amp 0 \amp 2 \\ -2 \amp 2 \amp -4 \amp -2 \\ \end{array}\right] \sim \left[\begin{array}{rrrr} 1 \amp 0 \amp 0 \amp 2 \\ 0 \amp 1 \amp -2 \amp 1 \\ 0 \amp 0 \amp 0 \amp 0 \\ \end{array}\right] \text{.} \end{equation*}
  1. Are the columns of \(A\) linearly independent? Do they span \(\real^3\text{?}\)

  2. Give a parametric description of the solution space to the homogeneous equation \(A\xvec = \zerovec\text{.}\)

  3. Explain how this parametric description produces two vectors \(\wvec_1\) and \(\wvec_2\) whose span is the solution space to the equation \(A\xvec = \zerovec\text{.}\)

  4. What can you say about the linear independence of the set of vectors \(\wvec_1\) and \(\wvec_2\text{?}\)

  5. Let's denote the columns of \(A\) as \(\vvec_1\text{,}\) \(\vvec_2\text{,}\) \(\vvec_3\text{,}\) and \(\vvec_4\text{.}\) Explain why \(\vvec_3\) and \(\vvec_4\) can be written as linear combinations of \(\vvec_1\) and \(\vvec_2\text{.}\)

  6. Explain why \(\vvec_1\) and \(\vvec_2\) are linearly independent and \(\span{\vvec_1,\vvec_2} = \span{\vvec_1, \vvec_2, \vvec_3, \vvec_4}\text{.}\)

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