###### Exercise10

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

1. What does this say about the pivots of the matrix $$A\text{?}$$ Write the reduced row echelon form of $$A\text{.}$$

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 $$A\xvec = \zerovec\text{?}$$

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

in-context