ekiim's blog

Exercises

1. Apply elimination and back-substitution to solve $2u -3v = 3 2u -5v + w = 7 2u - v -3w = 5${.aligned} Where are the pivots? List the three operations in which a multiple of one row is substracted from another.

2. For the system $u + v + w = 2 y + 3v + 3w = 0 y + 3v + 5w = 2,${.aligned} what is the triangular system after forward elimination, and what is the solution?

3. Solve the system and find the pivots when $2u - v = 0 -u + 2v - w = 0 - v +2w - z = 0 - w +2z = 5${.aligned} You may carry the right side as fifth column (and omit wrigint $u, v, w, z$ until the solution at the end).

4. Apply Elimination to the system$u + v + w = -2 3u + 3v - w = 6 y - v + w = -1${.aligned} When a zaero arises in the pivot position, exchange that equation for the one below it and proceed. What coefficient of $v$ in the third equation, in place of the present $-1$, would make it impossible to procceed — and force elimination to breakd down?

5. Solve by elimination the system of two equations $x - y = 0 3x + 6y = 18${.aligned} Draw a graph representing each equation as straight lines in the $x-y$ plane; the lines intersect at the solution. Also add one more line – the graph of the new equation which arises after elimination.

6. Find three values of $a$ for which elimination breaks down, temporarly or permanently, in $au + v = 1 4u + av = 2${.aligned} Breakdown at the first step can be fixed by exchanging rows – but not breakdown at last step.

1. If the first two rows of $A$ are the same, when will elimination discover that $A$ is singular? Do a 3 by 3 example, allowing row exchanges.
2. If the first two columns of $A$ are the same, when will elimination discover that $A$ is singular? Do a 3 by 3 example, allowing column exchanges.
7. How many multiplications-substractions would it take to solve a system of order $n = 600$?. How Many seconds, on a PC that can do $8000$ a seconds or a VAX that can do $80,000$, or a CRAY X-MP/2 that can do 12 million? (Thoes are in double precision — I think the CRAY is cheapest if you can afford it.)

8. True or false:

1. If the third equation starts with a zero coefficient (it begins with $0u$) then no multiple of equation 1 will be substracted from equation 3.
2. If the third equation has a zero in the second coefficient (it contains (0v) then no multiple of equation 2 will be substracted from equation 3.
3. If the third equation contains $0u$ and $0v$, then no multiple of equation 1, or equation 2 will be substracted from equestion 3.
9. (very optional) Normally the multiplitaction of two complex numbers $(a + ib)(c + id) = (ac - bd) + i(bc + ad)$ involves the four seperated multiplications $ac, bd, bc, ad$. Ingoring i, can you compute the quantities $ac-bd$ and $bc+ad$ with only three multiplications? (You may do additions, such as forming $a, b$, before multiplying without any penalty. We note hoever that additions take six cloc cycles on a CRAY X-MP/48 and multiplications is only one more time.)

10. Use elimination to solve

 \begin{aligned} u + v + w &= b_1 \\ v + w &= b_2 \\ w &= b_3 \end{aligned} And \begin{aligned} u + v + w &= 7 \\ u + 2v + 2w &= 10 \\ 2u + 2v - 4w &= 3 \\ \end{aligned}

The final exercises gives experience in setting up linear equations. suppose that (a) of those start a year in California, 80 percent stay in and 20 percent move out; (b) of those start a year in California, 90 percent stay in and 10 percent move out; If we know the situation at the beginning, say 200 million outside and 30 million in, the it is easy to find the number $u$ and $v$ that are outisde and inside at the end: \begin{aligned} .9(200,000,000) + .2(30,000,000) &= u\\ .1(200,000,000) + .8(30,000,000) &= v \end{aligned} The real problem is to go backwards, and compute the start from the finish.

1. If $u =$ 200 million and $v =$ 30 million at the end, set up (without solving) the equations to find $u$, and $v$, at the beginning.
2. If $u$ and $v$ at the end are teh same as $u$ and $v$ at the beginning what equations do you get? What is the ratio of $u$ to $v$ in this “steady state”?