Activity3.5.4

We will explore some column spaces in this activity.

  1. Consider the matrix

    \begin{equation*} A= \left[\begin{array}{rrr} \vvec_1 \amp \vvec_2 \amp \vvec_3 \end{array}\right] = \left[\begin{array}{rrr} 1 \amp 3 \amp -1 \\ -2 \amp 0 \amp -4 \\ 1 \amp 2 \amp 0 \\ \end{array}\right] \text{.} \end{equation*}

    Since \(\col(A)\) is the span of the columns, the vectors \(\vvec_1\text{,}\) \(\vvec_2\text{,}\) and \(\vvec_3\) naturally span \(\col(A)\text{.}\) Are these vectors linearly independent?

  2. Show that \(\vvec_3\) can be written as a linear combination of \(\vvec_1\) and \(\vvec_2\text{.}\) Then explain why \(\col(A)=\span{\vvec_1,\vvec_2}\text{.}\)

  3. Explain why the vectors \(\vvec_1\) and \(\vvec_2\) form a basis for \(\col(A)\text{.}\) This shows that \(\col(A)\) is a 2-dimensional subspace of \(\real^2\) and is therefore a plane.

  4. Now consider the matrix \(A\) and its reduced row echelon form:

    \begin{equation*} A = \left[\begin{array}{rrrr} -2 \amp -4 \amp 0 \amp 6 \\ 1 \amp 2 \amp 0 \amp -3 \\ \end{array}\right] \sim \left[\begin{array}{rrrr} 1 \amp 2 \amp 0 \amp -3 \\ 0 \amp 0 \amp 0 \amp 0 \\ \end{array}\right] \text{.} \end{equation*}

    We will call the columns \(\vvec_1\text{,}\) \(\vvec_2\text{,}\) \(\vvec_3\text{,}\) and \(\vvec_4\text{.}\) Explain why \(\vvec_2\text{,}\) \(\vvec_3\text{,}\) and \(\vvec_4\) can be written as a linear combination of \(\vvec_1\text{.}\)

  5. Explain why \(\col(A)\) is a 1-dimensional subspace of \(\real^2\) and is therefore a line.

  6. What is the relationship between the dimension \(\dim~\col(A)\) and the rank \(\rank(A)\text{?}\)

  7. What is the relationship between the dimension of the column space \(\col(A)\) and the null space \(\nul(A)\text{?}\)

  8. If \(A\) is an invertible \(9\times9\) matrix, what can you say about the column space \(\col(A)\text{?}\)

  9. If \(\col(A)=\{\zerovec\}\text{,}\) what can you say about the matrix \(A\text{?}\)

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