Di¤erential Equations and Matrix Algebra I (MA 221), Fall Quarter, 1999-2000
Quiz 4 –due Monday, September 20, 1999
INSTRUCTIONS: You are to work in groups of 2 - 4 people (no working alone). There is
no time limit for taking this quiz. Hand in one set of solutions per group (be sure to put all
names at the top of the …rst page).
1) Give the geometric picture (ie two 3–spaces with the null space of A on the left and
the range of A on the0right with an1arrow from left to right) for the system of equations
3 ¡2 0
¡
!
B
¡
!
A x = b where A = @ 1 1 5 C
A
2 5 19
0
1
1 ¡2 ¡1
3 C
2) Let A = B
@ 1 2
A : Find a basis for the column space of A in three di¤erent ways:
3 4
7
(a) by hand (ie by observation), (b) using Maple, and (c) using Matlab (here you will need
to enter the matrix and then type orth(A)).Show that all three bases give the same space
(this means that each basis vector must be in the span of the other bases). Also determine
the special properties of the basis which you get with Matlab.
Note that it is possible to …nd tutorials on Matlab on the web. For example, I recently saw
one at
http://www.math.mtu.edu/~msgocken/intro/intro.html
or go to Professor Leader’s web page
http://www.rose-hulman.edu/~leader/ma371mat.html
3) Set the following up as a system of equations (in fact, put the system in matrix form) and
solve.
4) Determine the ‡ow rates (cfs=cubic feet per second) for the following pipe system. Notice
that there are 5 unknown ‡ow rates (Q1 : : : Q5 ) and there will be 4 equations. That is, at
each node, the ‡ow in will equal the ‡ow out. First write down the four equations, put into
standard form, and then solve. You will be able to see from the mathematical solution that
there is not a unique solution (ie express your answer in vector parametric form). Try to
explain from a physical point of view why there is not a unique solution.
5) Set up the system of equations which will determine the positions of the bars when the
following system is in equilibrium. Solve using Matlab (give me a printout or write down
the Matlab commands that you used).