6 Linear Algebra III

1. Given
   
   $$A = \left( \begin{array}{rrrr} 8 & 2 & 1 & 0 \\ 0 & 1 & 1 & 3 \\ 4 & 0 & 0 & -3 \end{array} \right)$$
   
   
   Reduce to row canonical form using the class procedure (i.e. first to echelon avoiding fractions). At the same time find the matrix $\Omega$ so that $A_{\rm R}=\Omega A$. Is $\Omega=A^{-1}$? If not, why not? Is $A_{\rm R}=I$? What is the rank of $A$? What is its nullspace? What is the dimension of the null space? What is the solution space if the right hand side vector is $(1,2,3)$? Do you get the same solution space from the echelon and row canonical forms? Do you really need to include this right hand side in the augmented matrix or is there a simpler way?

2. Here are some quick ones. As always, explain all answers fully.
   1. Is an upper triangular matrix always in echelon form?
   2. Is a nonsingular upper triangular matrix always in echelon form?
   3. What is the null space of an $m\times n$ zero matrix? What is its dimension? What is the rank of the matrix?
   4. What is the null space of a unit matrix? What is the rank of the matrix?
   5. What is the dimension of the null space of an $1\times n$ nonzero matrix (i.e. a nonzero row vector)? Give it for matrix $(0, 0, 3, 0, 0, 6, 0)$. What is the rank of the matrix?
   6. Can a system $A_{m\times n} \vec x = \vec 0$ where $m>n$ (more equations than unknowns) have a nontrivial solution?
   7. Must there always be a solution to $A_{m\times n} \vec x = \vec b$ where $n>m$ (more unknowns than equations)?
   8. Prove that a system $A_{m\times n} \vec x = \vec 0$ with $m<n$ always has a nontrivial solution. So what can you say about the dimension of the null space?