Exercise 5

Determine whether the following statements are true or false and provide a justification for your response. Throughout, we will assume that the matrix \(A\) has columns \(\vvec_1,\vvec_2,\ldots,\vvec_n\text{;}\) that is,

\begin{equation*} A = \left[\begin{array}{rrrr} \vvec_1\amp\vvec_2\amp\ldots\amp\vvec_n \end{array}\right]\text{.} \end{equation*}
  1. If the equation \(A\xvec = \bvec\) is consistent, then \(\bvec\) is in \(\span{\vvec_1,\vvec_2,\ldots,\vvec_n}\text{.}\)

  2. The equation \(A\xvec = \vvec_1\) is always consistent.

  3. If \(\vvec_1\text{,}\) \(\vvec_2\text{,}\) \(\vvec_3\text{,}\) and \(\vvec_4\) are vectors in \(\real^3\text{,}\) then their span is \(\real^3\text{.}\)

  4. If \(\bvec\) can be expressed as a linear combination of \(\vvec_1, \vvec_2,\ldots,\vvec_n\text{,}\) then \(\bvec\) is in \(\span{\vvec_1,\vvec_2,\ldots,\vvec_n}\text{.}\)

  5. If \(A\) is a \(8032\times 427\) matrix, then the span of the columns of \(A\) is a set of vectors in \(\real^{427}\text{.}\)