Exercise3

Answer 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 \(A\text{.}\) If the equation \(A\xvec = \bvec\) is inconsistent, what can you say about the linear independence or dependence of the set of vectors \(\vvec_1,\vvec_2,\ldots,\vvec_n,\bvec\text{?}\)

in-context