Activity 2.4.3

Explain why the homogeneous equation $A\xvec = \zerovec$ is consistent no matter the matrix $A$.

Consider the matrix $$A = \left[\begin{array}{rrr} 3 & 2 & 0 \\ -1 & 0 & -2 \\ 2 & 1 & 1 \end{array}\right]$$ whose columns we denote by $\vvec_1$, $\vvec_2$, and $\vvec_3$. Describe the solution space of the homogeneous equation $A\xvec = \zerovec$ using a parametric description, if appropriate.

Find a nonzero solution to the homogeneous equation and use it to find weights $c_1$, $c_2$, and $c_3$ such that $$c_1\vvec_1 + c_2\vvec_2 + c_3\vvec_3 = \zerovec$$.

Use the equation you found in the previous part to write one of the vectors as a linear combination of the others.

Are the vectors $\vvec_1$, $\vvec_2$, and $\vvec_3$ linearly dependent or independent?