Description

13 attachmentsSlide 1 of 13attachment_1attachment_1attachment_2attachment_2attachment_3attachment_3attachment_4attachment_4attachment_5attachment_5attachment_6attachment_6attachment_7attachment_7attachment_8attachment_8attachment_9attachment_9attachment_10attachment_10attachment_11attachment_11attachment_12attachment_12attachment_13attachment_13

Unformatted Attachment Preview

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

Purchase answer to see full

attachment

Tags:

math

algebra

User generated content is uploaded by users for the purposes of learning and should be used following Studypool’s honor code & terms of service.

## Reviews, comments, and love from our customers and community:

This page is having a slideshow that uses Javascript. Your browser either doesn't support Javascript or you have it turned off. To see this page as it is meant to appear please use a Javascript enabled browser.