Exercise4

Determine if the following statements are true or false and provide a justification for your response.

  1. If \(\vvec_1,\vvec_2,\ldots,\vvec_n\) are linearly dependent, then one vector is a scalar multiple of the others.

  2. If \(\vvec_1, \vvec_2, \ldots, \vvec_{10}\) are vectors in \(\real^5\text{,}\) then the set of vectors is linearly dependent.

  3. If \(\vvec_1, \vvec_2, \ldots, \vvec_{5}\) are vectors in \(\real^{10}\text{,}\) then the set of vectors is linearly independent.

  4. Suppose we have a set of vectors \(\vvec_1,\vvec_2,\ldots,\vvec_n\) and that \(\vvec_2\) is a scalar multiple of \(\vvec_1\text{.}\) Then the set is linearly dependent.

  5. Suppose that \(\vvec_1,\vvec_2,\ldots,\vvec_n\) are linearly independent and form the columns of a matrix \(A\text{.}\) If \(A\xvec = \bvec\) is consistent, then there is exactly one solution.

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