Exercise 6

Suppose that $\vvec_1,\vvec_2,\ldots,\vvec_n$ is a set of vectors in $\real^{27}$ that form the columns of a matrix $A$.

1. Suppose that the vectors span $\real^{27}$. What can you say about the number of vectors $n$ in this set?

2. Suppose instead that the vectors are linearly independent. What can you say about the number of vectors $n$ in this set?

3. Suppose that the vectors are both linearly independent and span $\real^{27}$. What can you say about the number of vectors in the set?

4. Assume that the vectors are both linearly independent and span $\real^{27}$. Given a vector $\bvec$ in $\real^{27}$, what can you say about the solution space to the equation $A\xvec = \bvec$?