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 $A$.

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

3. If a linear system of equations has 8 equations and 5 unknowns, then the shape of the matrix $A$ in the corresponding equation $A\xvec = \bvec$ is $5\times8$.

4. If $A$ has a pivot position 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 $\bvec$.