Exercise 1

Consider the set of vectors $$\vvec_1 = \threevec{1}{2}{1}, \vvec_2 = \threevec{0}{1}{3}, \vvec_3 = \threevec{2}{3}{-1}, \vvec_4 = \threevec{-2}{4}{-1}$$.

1. Explain why this set of vectors is linearly dependent.

2. Write one of the vectors as a linear combination of the others.

3. Find weights $c_1$, $c_2$, $c_3$, and $c_4$, not all of which are zero, such that $$c_1\vvec_1 + c_2 \vvec_2 + c_3 \vvec_3 + c_4 \vvec_4 = \zerovec$$.

4. Suppose $A=\left[\begin{array}{rrrr} \vvec_1&\vvec_2&\vvec_3&\vvec_4 \end{array}\right]$. Find a nonzero solution to the homogenous equation $A\xvec = \zerovec$.