Exercise6

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

  1. Suppose that the vectors span \(\real^{27}\text{.}\) 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 \(\real^{27}\text{.}\) What can you say about the number of vectors in the set?

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

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