MATH 410 American River College Linear Algebra Questions

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Math 410 Take Home Exam 1 (Ch.1, 2.1)
Exam Instructions:
Use your own paper and hand write the solutions to each of the problems. You may use a tablet if you
wish. Put your name on each page . . just in case. Each problem should be completed on a its own sheet
or sheets of paper
When finished you will scan your exam solutions to PDF and submit it just like HW.
Show all work. Problems with an answer and no work will be give a zero, regardless of correctness.
You must show all your setup work. I want to see the integrals fully worked out by hand unless
otherwise stated in the problem.
You must work alone. No working in groups. No help from tutors or others. It must be your own
work.
If your work matches one or more person’s work, then ALL people with the matching work will
be given a 0 on that problem. Both the person copying and the person who allowed their work to
be copied will be penalized equally.
You may use the textbook, notes and anything from our class Canvas site.
You may use a graphing calculator except on problems that specifically ask that you solve by hand.
You may not make use of Wolfram Alpha, or any other online linear algebra resource to do your work
for you. If your work looks like it came from Wolfram Alpha, I’ll run the problem thru Wolfram myself.
If your work matches the output of Wolfram Alpha, the problem will be graded as a 0.
Exam is due at Mon, Oct5 @ 3:30pm sharp. The exam submission will close exactly at 3:30pm
and I will not accept late exams for any reason.
If you don’t get it in by 3:30pm, its a 0, no exceptions. You are being given an extremely liberal
5 day window to work on the exam. Plan to finish the exam by Mon, 9/28 @ Noon to give
yourself plenty of time to submit it. This way if anything goes wrong, or you have internet
problems, you can still contact me using your phone and we can work things out.
I will be available in office hours on Mon, 10/5 from 10:30 to Noon and via email from Noon
until 12:45p in the event you have a tech problem related to submission.
If you wait till the last minute to finish your exam and submit it and then miss the deadline, that’s
on you . . your fault and no one else’s. Seriously, plan to finish by Mon @ Noon so you have
time to deal with any problems. If you are finished earlier (Sat or Sun say) you are welcome to
turn it in earlier.
Check your file before you submit it! If pages are missing, those problems are a 0. If pages
are unreadable, those problems are a 0. Its your exam, your grade . . its your responsibility
to make sure the file is correct.
1. (6pts) Solve the following linear system of equations by hand. Show all your work.
2. (5pts) Use matrices to determine whether the following system of linear equations is consistent or
inconsistent and why. You may use a calculator but show your setup and results from your calculator.
3. For
C = [1 -1 -2]
Perform the following matrix operations by hand.
If an operation is not possible, write “Not possible” and explain why
a) (3pts)
b) (3pts)
c) (3pts)
4. (5pts) Solve the following system by hand and put your solution in parametric vector form
5. Given
vectors
and
a) (4pts) Is c a linear combination of
b) (4pts) Is c in span
c) (6pts) Do
? Show your work to justify your answer
? Why? Show your work to justify your answer
span all of
?
6. (6pts) Solve the matrix equation
for .
Write your answer as a column vector. You may use a calculator for this but show your problem setup and any
entries into your calculator as part of your work.
7. For
defined by
a) (4pts) Is b =
T( ) =
in Range(T)?
b) (4pts) Is T a linear transformation? Why or why not?
c) (4pts) Is T onto
? Show work to justify your answer
d) (4pts) Is T a 1-1 transformation? Show work to justify your answer
8. (7pts) Is
a linear transformation? Use the definition of linear
transformation to show why it is or is not
9. (7pts) The projection of vectors from
into
is a linear transformation defined by
.
Determine the standard matrix A for this transformation so that T(x) = Ax.
10. If
and
a) (2pts) Confirm that
is a solution to the non-homogeneous system Ax=b
b) (2pts) Confirm that
is a solution to the homogeneous system Ay=0
c) (3pts) Will any scalar multiple of y solve the homogeneous system Ay=0? Use properties of scalar
multiplication and matrix multiplication to show why or why not.
d) (3pts) Write the general solution to the non-homogeneous system Ax=b using parametric form
11. Determine by inspection whether the following sets of vectors are linearly dependent or linearly
independent and justify your answer by describing why
a) (3pts)
b) (3pts)
c) (3pts)
12. (6pts) Do the columns following matrix A form a linearly independent or linearly dependent set of vectors?
Justify your answer by showing all you work and showing any calculations you make or matrices you use to
come to that conclusion.
Uses for Systems of Equations and Matrices
Systems of Equations, naturally arise from many different situations in the world around us.
“Real World” problems tend to have many different variables. Because of this, the systems of
equations that we look at in the world around us tend to be on the larger size. Large systems are
best solved using matrices and techniques we’ve seen so far in class.
Below we’ll walk through a few different scenarios where system of equations arise and we’ll
work with the resulting systems and matrices.
Scenario 1: Network Flow
One unique situation where matrices and systems arise is when studying network flow. This is
the study of the flow of ‘something’ across a grid (the network) which has varying speeds and
constraints.
A good example of what is meant by network flow can be seen on p52, Example 2 of our
textbook. We have different paths in a grid which define the directions and capacity of the flow.
This grid could represent city street if you are urban planner, it could represent nodes in a data
network if you are a computer network engineer, it could represent transportation routes
between cities or countries if you are an economist or it could water flow across a city water
network if you are a civil engineer (just to name a few possibilities).
The “something” that is flowing could corresponding be car traffic, data, goods being shipped
along transportation routes or water flowing thru a water/sewer network.
In Example 2, we have traffic flow across city streets in an urban center.
The streets are 1 way streets with intersections labeled A, B, C, D.
Numbers at the far ends of the grid represent traffic flow in vehicles per hr.
Variables are attached to city streets to describe the flow along each of the those streets.
(see p52 picture)
Goals:
Find the conditions where traffic flows “nicely” with no bottlenecks.
Look at the effects that will occur if a street is closed due to repair, accidents, etc.
Assumptions:
If traffic flows without bottlenecks, then we can assume, at each intersection,
the flow of traffic into the intersection is equal to the flow out of the
intersection. This gives us a starting point.
Take each intersection A, B, C, D and associate an equation flow in = flow out with it.
A:
300 cars/hr coming in via Pratt St., 500 car/hr come in from the south
Some unknown number of the those cars exit the intersection to the North,
and some unknown number exit the intersection to the East,
A:
B:
C:
D:
Similarly we can get equations for intersections B,C,D
Moving all the variables to the left side and all the constants to the right, we get the following
system
Solving this system of equations will give flows along each street such that no bottlenecks occur.
To solve this, we notice a couple of things. 1st . . there are more variable than equations. This
system will likely be a dependent system. So traffic flow along some streets will depend on
traffic flow along one or others (no big surprise). 2nd, this system will be easiest to solve with
augmented matrices.
In matrix form we get
row reducing this via calculator leads to
Interpreting this
So we get that
is a free variable. There are some
constraints however. Traffic flows must be positive. So
If
in Eq1,
. If
in Eq4,
.
We can conclude that we need
to ensure
steady traffic flow with the numbers we see.
If
, then we can calculate what traffic all the other city streets would bear to see the
give outflow values (300 cars/hr on Lombard St, 600/hr East of D, etc)
From a planning perspective, we can also see the impacts of closures/accident.
What if
was closed?
Its flow = 0. Then
,
,
,
What if there was an accident closing ?
Big problems b/c now
so
which can’t happen.
So our grid can’t handle traffic in this way an still maintain the given outflows
What if we closed ?
traffic would jump, and therefore less of
traffic could go toward D else it’d
bottleneck so more of traffic at A would need to go North toward B.
would see heavy traffic (
=700) assuming it could handle that volume of cars
Network Flow Practice Problems (on HW assignment): p55: 14, 15
Scenario 2: Electrical Flow through Resistors (Kirchhoff’s Law)
Another place where matrices and linear algebra show up is when working with electricity and
circuits. These are extensive topics that you’d cover in depth in Physics 421 so we’re going to
just look at a limited case and lean on a couple of physical laws related to electricity.
First, a circuit in electricity is a closed loop which has a voltage source within the loop (like a
battery). This loop can have switches that break the loop or close the loop (like a light switch). It
can also have resistors, which impede the flow of electrons along the circuit or capacitors, which
can temporarily store electrical charge.
Looking on p82, Figure 1 and example 2, circuits are often drawn as shown. The saw tooth parts
represent resistors, the longer vertical line with shorter bold vertical line represents a voltage
source and the arrow represents how the current will flow thru the loop.
When working with circuits and resistors, there are 3 quantities that we are concerned with, the
voltage V (measured in volts), the amount of current flowing I (measured in amps) and the
amount of resistance R at different points, (measured in ohms).
As electricity flows through the circuit, the resistors will cause the voltage to drop. The amount
of the drop is given by Ohm’s Law which is
Each time you hit a resistor in the loop, you
get a voltage drop.
If there are multiple resistors in the loop, Kirchhoff’s Law says that sum of voltage drops in one
direction around the loop equals the sum of the voltage sources directly connected to the loop.
Lastly, current can be induced to flow in either direction of the loop. If the current starts at the
longer (positive end) of the voltage source and finished at the shorter, bold end of the voltage
source, the voltage is said to be positive. If the current starts at the shorter, bold (negative) end
of the voltage source, voltage said to be negative.
Now, looking at Figure1, there are 3 loops defined. In the upper loop there is 1 30V voltage
source. There are 3 resistors in this loop with resistance 4ohms, 3ohms and 4ohms each. The
arrow tells us the current is flowing counterclockwise, meaning it starts on the long side of the
voltage source and it finishes on the short,bold side. So the current is flowing from positive to
negative so voltage is positive.
For loop 1, we don’t know the current
.
but we do know, by Ohm’s Law the voltage drops will be
If there were no other loops in our circuit, with Kirchhoff’s Law, we could say
and solve
for the current. But, there is an adjoining 2nd loop and they share a wire connecting A + B. So
current from 2nd loop is going to impact loop 1.
Looking at loop 2, the voltage drops, starting at the 5V voltage source are
.
Current along loop 2 is also flowing counterclockwise. It leaves the voltage source at the longer
end so this means that the voltage is positive.
Going back to loop 1, with Kirchhoff’s Law and taking into account the overlapping segment from
loop2, we get
.
Simplifying this we get an equation
Note: Since current from the 2nd loop is going across wire AB in a different direction from loop 1,
it has a different sign than the others.
For loop 2, we have 2 wires that are shared, AB is shared between loop1 and loop2 and CD is
shared between loop 2 and loop 3.
For loop 2, the voltage drops were
but we also need to take into account
current from loops 1 and 3. We already saw that current across AB in loop1 will be
and its in
the opposite direction from loop2. So its sign will be opposite the loop2 sign
So we have
As far as loop 3, the shared wire CD has a voltage drop of
opposite across wire CD the sign of
will be negative.
and since current is also flowing
Therefore for loop2 we have
so
Note: Voltage source for loop 2 is positive again so sum of voltage drops = 5
Lastly for loop 3 we get
for the voltage drops but it shares a wire with loop 2 so we
also have
. In loop 3 there are 2 voltage sources and since current flows counter clockwise,
it is leaving the bold end of the source so it is a negative voltage.
That means, via Kirchhoff’s Law, we have
or
Putting all 3 loops together we have a system of equations
This can be solved with augmented matrices to get
=3amps
=1amp
=-8amps
Key things with circuits are
(1) you have to pay attention to whether the voltage source is positive or negative. That
is based on both the direction of flow (clockwise vs. counterclockwise) AND as a
result, is the flow starting at the positive end of the voltage source (long vert line) or
the negative end (bold, short line)
(2) When loops overlap, you have to take into account voltage drops from both loops along
the overlapping segment. Signs of those drops will be opposite each other.
Electrical Flow Practice Problem (on HW assignment): p86 #7
Scenario 3: City Migration
A last example of the use of matrices and systems of equations is working with demographics.
Looking at Folsom and Sacramento, each year a certain percentage of people living in Folsom
move to Sacramento (closer to work, want to live in the city, etc).
Conversely, a certain percentage of people living in Sacramento move to “the suburbs” of
Folsom. (more space, better schools, etc).
For the sake of example, lets assume these percentages are fixed. Each year 1% of Sacramento
residents move to Folsom and the rest (99%) stay in Sacramento. Similarly, each year 5% of
Folsom residents move to Sacramento and the rest (95%) stay in Folsom. We’ll assume that no
other factors affect the populations of Folsom and Sacramento.
We also know, from census data that the population of Folsom in 2010 was 72,000 and the
population of Sacramento in 2010 was 466,000.
Now, lets call the population of Folsom in 2010
and the population of Sacramento in 2010
So,
,
. Further we’ll say
= Folsom population in 2010 + i
= Sacramento population in 2010 + i
Now given our assumptions about migration rates
= Folsom population in 2011 =
= Sacramento population in 2011 =
These equations above are called difference equations. This system of difference equations can
be expressed in matrix/vector form
and calling
,
and A =
we get
where
is the 2011 population vector and
population vector
is 2010
Since we’re assuming the migration rates between Folsom and Sacramento don’t change, we can
say that
etc
We can easily see in general
where
is population vector for 2010 + i
Expressed this way, we have a succinct formula to find the population vector (which is both
Folsom and Sacramento’s population in any given year.
For example, to find the populations in 2020, we want
So based on our assumptions, we would
expect the population of Folsom to be 80,151
and Sacramento to be 457,849 this year.
A final planning question to ask would be, do the populations of Sacramento and Folsom ever
stabilize?
This equivalent to asking, what is the long term trend of these populations?
Expressed mathematically, we want to know
and
Limits for powers of matrices is something that is beyond our ability at the moment but will be a
problem we will come back and tackle toward the end of this course.
City Migration Practice Problems (on HW assignment): p87 9,11

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