Exercise 10

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.