Exercise 3

Respond to the following questions and provide a justification for your responses.

1. If the vectors $\vvec_1$ and $\vvec_2$ form a linearly dependent set, must one vector be a scalar multiple of the other?

2. Suppose that $\vvec_1,\vvec_2,\ldots,\vvec_n$ is a linearly independent set of vectors. What can you say about the linear independence or dependence of a subset of these vectors?

3. Suppose $\vvec_1,\vvec_2,\ldots,\vvec_n$ is a linearly independent set of vectors that form the columns of a matrix $A$. If the equation $A\xvec = \bvec$ is inconsistent, what can you say about the linear independence or dependence of the set of vectors $\vvec_1,\vvec_2,\ldots,\vvec_n,\bvec$?