Exercise 4

Determine whether 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 one of the others.

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

3. If $\vvec_1, \vvec_2, \ldots, \vvec_{5}$ are vectors in $\real^{10}$, 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 $\vvec_1$. 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 $A$. If $A\xvec = \bvec$ is consistent, then there is exactly one solution.