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) UNGRADED: 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) UNGRADED: 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) UNGRADED: Repeat for $\vec b^{ \rm T} = (0,-3,3)$. (f) Find the rank of $A$. (g) UNGRADED: 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) UNGRADED: 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. UNGRADED: 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.