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. Consider the system of equations
   $$\begin{eqnarray*} \phantom{-}4 x_1 - \phantom{0}x_2 + 4 x_3 & = & 1 \\ \phant... ...2 - 5 x_3 & = & 0 \\ - 2 x_1 + \phantom{0}x_2 + 7 x_3 & = & 4 \end{eqnarray*}$$
   
   
   Solve this system, as written (not in matrix notation), using the class procedure

   In the first stage of the forward elimination, you will first need to do a partial pivoting to avoid fractions. In the second stage, partial pivoting does no good, so should not be done. Since you cannot take an integer multiple of the original equation different from 1 in this question (since it would mess up the $L$ matrix asked later) you will need to live with fractions in the final equation.

   Solve the resulting equations using the class procedure.

   Next do the same but using augmented matrix notation. (See also the revised notes on linear algebra.)

   Next find the $L$ and $U$ matrices of the $LU$ decomposition and multiply the result. Do they give back the original matrix $A$ of the system. If not, what do they give back? So, what is the right hand vector $\vec b^{\rm pp}$ to use when solving $L\vec b^*=\vec b^{\rm pp}$? Solve this system and check that you get indeed the correct right hand side vector to use in solving the system $U\vec x=\vec b^*$. Double-check you are using the correct matrix and right hand side while doing all the above. Make sure you have answered everything. [2, 8.1].

2. 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 $\vec b^{ \rm T} = (0,-3,3)$. [Continued next homework.]

3. 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?