###### Activity2.4.3Linear independence and homogeneous equations
1. Explain why the homogenous equation $$A\xvec = \zerovec$$ is consistent no matter the matrix $$A\text{.}$$

2. Consider the matrix

\begin{equation*} A = \left[\begin{array}{rrr} 3 \amp 2 \amp 0 \\ -1 \amp 0 \amp -2 \\ 2 \amp 1 \amp 1 \end{array}\right] \end{equation*}

whose columns we denote by $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ and $$\vvec_3\text{.}$$ Are the vectors $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ and $$\vvec_3$$ linearly dependent or independent?

3. Give a description of the solution space of the homogeneous equation $$A\xvec = \zerovec\text{.}$$

4. We know that $$\zerovec$$ is a solution to the homogeneous equation. Find another solution that is different from $$\zerovec\text{.}$$ Use your solution to find weights $$c_1\text{,}$$ $$c_2\text{,}$$ and $$c_3$$ such that

\begin{equation*} c_1\vvec_1 + c_2\vvec_2 + c_3\vvec_3 = \zerovec\text{.} \end{equation*}
5. Use the expression you found in the previous part to write one of the vectors as a linear combination of the others.

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