6 Linear Algebra III

In this class,

• Questions must be answered in order asked.
• Solutions must be neat.
• Solutions must be exact and fully simplified.
• You must use the given symbols.
• You must show all reasoning.
• Copying is never allowed, even when working together.

1. Given
   
   $$A = \left( \begin{array}{rrrr} 6 & 4 & 2 & 2 \\ 9 & 6 & 0 & 0 \\ 3 & 2 & 4 & 4 \end{array} \right)$$
   
   
   Do all of the next things using the class procedures: (a) Reduce to echelon form. Avoid fractions in both echelon matrix and multipliers, but use only legal partial pivoting to achieve that. Do not use non-unit multiples of the rows being changed. Check your result carefully. (b) Using class procedures, find the null space of $A$. State its dimension. (c) Calling the echelon form $U$, find $L$ so that $LU=A^{\rm pp}$ is like $A$, but with permuted rows. (Note that the second pivoting will permute the elements in the first column of $L$; the first column should be 1,2,3.) (d) Use matrices $L$ and $U$ to quickly find the solution space, if any, for $A\vec x=\vec b$ if $\vec b^{ \rm T} = (0,1,0)$. (e) Repeat for $\vec b^{ \rm T} = (0,-3,3)$. (f) Find the rank of $A$. (g) Explain why the sum of the rank of $A$ and the dimension of the null space equals the number of columns of $A$. (h) What is the dimension of the row space of $A$? Find a fully simplified basis for it. Write the expression for the row space in terms of that basis. (i) Repeat the previous question for the column space. No, do not use $U$ here, that is wrong as the column space gets destroyed going from $A$ to $U$. (j) Neatly draw the two basis vectors of the column space in a three dimensional coordinate system and so illustrate a triangular piece of the row space plane.

2. Here are some quick ones. As always, explain all answers fully.
   1. Is an upper triangular matrix always in echelon form? Why?
   2. Is a nonsingular upper triangular square matrix always in echelon form? Why?
   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 its dimension? 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 the null space 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. Why does a system $A_{m\times n} \vec x = \vec 0$ with $m<n$ always have a nontrivial solution? So what can you say about the dimension of the null space?

3. Given
   
   $$A = \left( \begin{array}{rrrr} 4 & 3 & -5 & 6 \\ 1 & ... ... \\ 0 & -5 & 1 & 7 \\ 8 & 9 & 0 & 15 \end{array} \right)$$
   
   
   Find the determinant of this matrix using minors. Minimize the algebra in doing so. No Gaussian elimination steps, including partial pivoting, allowed.

4. Reconsider the matrix:
   
   $$A = \left( \begin{array}{rrrr} 4 & 3 & -5 & 6 \\ 1 & ... ... \\ 0 & -5 & 1 & 7 \\ 8 & 9 & 0 & 15 \end{array} \right)$$
   
   
   Find the determinant of this matrix using Gaussian elimination. Compare to your earlier result.