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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