Exercise 10

Suppose that $A$ is a $4\times4$ matrix and that the equation $A\xvec = \bvec$ has a unique solution for some vector $\bvec$.

1. What does this say about the pivot positions of the matrix $A$? Write the reduced row echelon form of $A$.

2. Can you find another vector $\cvec$ such that $A\xvec = \cvec$ is inconsistent?

3. What can you say about the solution space to the equation $A\xvec = \zerovec$?

4. Suppose $A=\left[\begin{array}{rrrr} \vvec_1 & \vvec_2 & \vvec_3 & \vvec_4 \end{array}\right]$. Explain why every four-dimensional vector can be written as a linear combination of the vectors $\vvec_1$, $\vvec_2$, $\vvec_3$, and $\vvec_4$ in exactly one way.