Exercise 5

Suppose we have a set of vectors $\vvec_1,\vvec_2,\vvec_3,\vvec_4$ in $\real^5$ that satisfy the relationship: $$2\vvec_1 - \vvec_2 + 3\vvec_3 + \vvec_4 = \zerovec$$ and suppose that $A$ is the matrix $A=\left[\begin{array}{rrrr} \vvec_1&\vvec_2&\vvec_3&\vvec_4 \end{array}\right]$.

1. Find a nonzero solution to the equation $A\xvec = \zerovec$.

2. Explain why the matrix $A$ has a column without a pivot position.

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

4. Explain why the set of vectors is linearly dependent.