Exercise 9

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

  1. If \(A\xvec\) is defined, then the number of components of \(\xvec\) equals the number of rows of \(A\text{.}\)

  2. The solution space to the equation \(A\xvec = \bvec\) is equivalent to the solution space to the linear system whose augmented matrix is \(\left[\begin{array}{r|r} A \amp \bvec \end{array}\right]\text{.}\)

  3. If a linear system of equations has 8 equations and 5 unknowns, then the dimensions of the matrix \(A\) in the corresponding equation \(A\xvec = \bvec\) is \(5\times8\text{.}\)

  4. If \(A\) has a pivot in every row, then every equation \(A\xvec = \bvec\) is consistent.

  5. If \(A\) is a \(9\times5\) matrix, then \(A\xvec=\bvec\) is inconsistent for some vector \(\bvec\text{.}\)