# SIT292 LINEAR ALGEBRA 2017

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** ****SIT292 LINEAR ALGEBRA 2017**

**Assignment 3**

**Due: 5 p.m. September 28, 2017**

Note that by University regulations, the assignment must reach the unit chair

by the due date, even if it is posted.

- State the de_nition of the row-rank. For the following matrix

*A *=

2

4

1 2 **3 0

2 4 **2 2

3 6 **4 3

3

5

(a) determine the row-rank.

(b) _nd a set of generators for the row space of *A*.

(c) _nd a basis for the row space of *A*. Explain why it is a basis.

( 4 + 2 + 4 = 10 marks)

- For the following matrix

2

664

0 2 0

1 0 1

0 2 0

3

775

(a) _nd the eigenvalues

(b) _nd the eigenvectors corresponding to these eigenvalues

(c) starting with the eigenvectors you found in (a) construct a set of

orthonormal vectors (use the Gram-Schmidt procedure).

( 5 + 10 + 5 = 20 marks)

- The set of ordered triples
*f*(1*;*0*;*1)*;*(**1*;*1*;*1)*;*(0*;*1*;*0)*g*forms a basis

for R3. Starting with this basis use the Gram-Schmidt procedure to

construct an orthonormal basis for R3.

( 10 marks)

- Denote by
*R**n*the set of all*n*-tuples of real numbers.*R**n*is called

the Euclidean vector space, with equality, addition and multiplication

de_ned in the obvious way. Let *V *be the set of all vectors in *R*4

orthogonal to the vector (0*; *1*;***2*; *1); i.e. all vectors *v **2 **V *so that

*v**T *(0*; *1*;***2*; *1) = 0.

(a) Prove that *V *is a subspace of *R*4.

(b) What is the dimension of *V *(provide an argument for this), and

_nd a basis of *V *. (Hint: observe that the vector (0*; *1*;***2*; *1)

does not belong to *V *, hence dim *V **_ *3; next _nd 3 linearly

independent vectors in *V *.)

(10 + 14 = 24 marks)

- Determine the dimension of the subspace of
*R*4 generated by the set of

4-tuples

*f*(1*; *2*; *1*; *2)*; *(2*; *4*; *3*; *5)*; *(3*; *6*; *4*; *9)*; *(1*; *2*; *4*; *3)*g*

(6 marks)

- The code words

*u*1 = 1010010*; u*2 = 1100001*; u*3 = 0101000*; u*4 = 0010100

form a basis for a (7*; *4) linear binary code.

(a) Write down a generator matrix for this code.

(b) Construct code words for the messages 1001 and 0101.

(c) Write down the parity check matrix for this code.

(d) Find the syndromes for the received words

1110011*; *1001010*; *0001101*; *1101010

(4 + 4 + 4 + 8 = 20 marks)

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# Need Assignment help-Math 261-01 Exam 2 EC ; http://customwritings-us.com/orders.php

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Math 261-01 Exam 2 EC

Due: 12:30 pm on March 14th

Name : _____________________________________________________________________________

This optional assignment could up to 10 points extra credit to your Exam 2 grade. Please show appropriate work and use terminology and notation correctly. Unless specified otherwise, you should give exact answers. Partial credit will be given sparingly, if at all. Each problem is worth 1 point. Remember that these should help you practice some of the concepts you will be tested over, but this assignment should not be interpreted as a sample exam.

1. Find the curvature of the curve 𝑟𝑟⃗(𝑡𝑡)=<4sin(2𝑡𝑡),4cos(2𝑡𝑡),4𝑡𝑡>.

2. At what point on the curve 𝑥𝑥=𝑡𝑡3,𝑦𝑦=5𝑡𝑡,𝑧𝑧=𝑡𝑡4 is the normal plane parallel to the plane 3𝑥𝑥+5𝑦𝑦−4𝑧𝑧=2?

3. Find the position vector of a particle that has the given acceleration and the given initial velocity and position:

𝑎𝑎⃗(𝑡𝑡)=sin(2𝑡𝑡),𝑣𝑣⃗(0)=1,𝑟𝑟⃗(0)=1

.

.

Name : ______________________________________________________________________

.. Page 2

4. (a) Evaluate the limit:

(b) Evaluate the limit: .

Name : ______________________________________________________________________

.. Page 3

5. Find for .𝑧𝑧=𝑦𝑦𝑦 𝑦 𝑦𝑦(6𝑥𝑥).

6. Find the equation of the tangent plane to the given surface at the specified point.

Name : ______________________________________________________________________

.. Page 4

7. Let and suppose that (x, y) changes from (2, –1) to (2.01, –0.98)

(a) Compute Δz. (b) Compute dz.

8. Use the Chain Rule to find .

.

Name : ______________________________________________________________________

.. Page 5

9. Find the directional derivative of the function 𝑓𝑓(𝑥𝑥,𝑦𝑦)=(𝑥𝑥+6)𝑒𝑒𝑦𝑦at the point P(3, 0) in the direction of the unit vector that makes the angle 𝜃𝜃=𝜋𝜋3 with the positive x-axis.

10. Find and classify the relative extrema and saddle points of the function

𝑓𝑓(𝑥𝑥,𝑦𝑦)=𝑥𝑥2+2𝑦𝑦2+𝑥𝑥2𝑦𝑦+11

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# MT560 Algebraic Topology Autumn 2016 – Homework 2

MT560 Algebraic Topology Autumn 2016 – Homework 2

1. Suppose that X,Y are spaces with Y Hausdorﬀ and f : X → Y is continuous. Show that the graph of f, Gr(f) = {(x,f(x)) ; x ∈ X}, is a closed subset of X ×Y .

2. Let (X,T) be a topological space and ‘∗’ a ‘point’ not in the set X. Let X0 = X ∪{∗}. (a) Let T0 = T ∪{X0}. Show that T0 is a topology on X0. (b) Show that X is a subspace of X0. (c) Show that (X0,T0) is compact. (d) Suppose X is Hausdorﬀ. Does it follow that X0 is Hausdorﬀ?

3. Alexandroﬀ 1-point compactiﬁcation. Let (X,T) be a topological space and ∗ a ‘point’ not in the set X. Let ˆ X = X ∪{∗}. (a) Let TA := T ∪{U ∪{∗} ; X − U is closed and compact in X}. Show that ( ˆ X,TA) is a compact topological space containing X asa subspace. (Note that X is open in ˆ X.) (b) Suppose that X is Hausdorﬀ and locally compact (i.e. for each x ∈ X, there exists an open set U such that x ∈ U, and its closure, ¯ U, is compact). Deduce that ˆ X is Hausdorﬀ. (c) Suppose X is not compact. Deduce that X is dense in ˆ X. Is this the case if X is compact? (d) Show that the Alexandroﬀ 1-point compactiﬁcation of the open n- cell en is homeomorphic to the n-sphere Sn. Show that attaching the n-disc Dn to a 0-cell ∗, via the constant map f : ∂Dn →{∗}, also gives the n-sphere.

# Need Help-SPSS Project

* Need Help-SPSS Project*

__Follow all instructions carefully __in presenting your answers. Be sure to __show all your working__. (Handwritten responses are fine.) You will not need SPSS for questions 1-3. For question 4, please download the housing dataset (from Latte), then import it into SPSS for analysis.

- Jet Blue Airlines examined the bags of 80 passengers and found that 20% of the bags were overweight.

- Based on this sample, what is the 95% confidence interval for the proportion of bags that are overweight? [6 points]

- What is the minimum sample size the airline would need to estimate with 95% confidence to obtain a margin of error of +/- 3% for this estimate of the percentage of overweight bags? [6 points]

- A factory recently took a sample to assess the quality of its candy output, looking at three different types of candy, and how many of each type of candy were damaged during the manufacturing process:

Candy |
# damaged |
Total # candies counted |

Apple hard candy | 15 | 50 |

Chocolate chew | 18 | 50 |

Nut cluster | 30 | 100 |

The factory management would like to determine whether the proportion of candy that is damaged is different for these three types of candy.

- Construct a contingency table for these data. [2 points]

- Is the proportion of candy that is damaged different for these three types of candy? (Calculate the appropriate statistic, give the p-value, and state your ) [6 points]

- A manufacturer of headphones is interested in the sales of a particular headphone model in its stores in 8 airports. Some of these stores are located on the West and some on the East coast of the U.S. Also, the manufacturer recently conducted an advertising campaign. The sales before and after the advertising campaign, which it ran in February using billboards in the airports, are shown below (i.e., data for sales in those stores in January and data for sales in the same stores for )

(Some descriptive statistics have also been provided in the table. You will need to decide which ones you need for your calculations in answering the questions below.)

Store |
Location |
Sales in Jan |
Sales in March |
Change in sales |

1 | East coast | 195 | 230 | 35 |

2 | East coast | 220 | 240 | 20 |

3 | East coast | 220 | 250 | 30 |

4 | East coast | 245 | 265 | 20 |

5 | West coast | 130 | 157 | 27 |

6 | West coast | 130 | 140 | 10 |

7 | West coast | 80 | 99 | 19 |

8 | West coast | 185 | 207 | 22 |

Summary statistics All stores |
||||

Mean | 175.63 | 198.50 | 22.88 | |

SD | 56.72 | 59.65 | 7.68 | |

East coast | ||||

Mean | 220.00 | 246.25 | 26.25 | |

SD | 20.41 | 14.93 | 7.50 | |

West coast | ||||

Mean | 131.25 | 150.75 | 19.50 | |

SD | 42.89 | 44.71 | 7.14 |

To get full points when answering __each part __below be sure to: calculate an appropriate statistic, state the result of the test, and state your conclusion.

- Looking at all the stores, is there a difference in sales between January and March? [6 points]
- Did the campaign have a different effect on sales for stores on the East coast versus on the West coast? [6 points]
- Was there a difference in sales
__in January__for stores on the East coast versus on the West coast? [6 points]

- Below are data for 40 houses located in one of two neighborhoods (A or B).

(This data is also provided in an Excel spreadsheet on the website for the class. Open the data in SPSS and conduct the analyses required to answer the questions. __Be sure to paste output (i.e., tables) from SPSS into your answers where that is requested __or else you will lose points.)

Neighborhood |
Appraised Land Value | Appraised Value of Improvements |
Sale Price |
Has a yard? (yes/no) |

A | 56658 | 53806 | 255000 | no |

A | 93200 | 11121 | 422000 | no |

A | 76125 | 78172 | 290000 | no |

A | 28996 | 5864 | 305900 | no |

A | 30000 | 64831 | 118500 | yes |

A | 30000 | 50765 | 93900 | yes |

A | 46651 | 8573 | 191500 | yes |

A | 45990 | 91402 | 184000 | yes |

A | 42394 | 98181 | 168000 | yes |

A | 47751 | 3351 | 169000 | yes |

A | 63596 | 2182 | 208500 | yes |

A | 51428 | 72451 | 264000 | yes |

A | 54360 | 61934 | 237000 | yes |

A | 65376 | 34458 | 286500 | yes |

A | 42400 | 15046 | 202500 | yes |

A | 40800 | 92606 | 168000 | yes |

A | 12170 | 22786 | 375000 | yes |

A | 24637 | 90598 | 169900 | yes |

A | 30600 | 80858 | 135000 | yes |

A | 44730 | 99047 | 176000 | yes |

B | 38979 | 25946 | 140000 | no |

B | 14861 | 59258 | 74900 | no |

B | 14976 | 48957 | 57300 | no |

B | 15244 | 55169 | 87500 | no |

B | 18260 | 59267 | 82000 | no |

B | 16680 | 55525 | 78000 | no |

B | 53421 | 19792 | 175000 | no |

B | 31417 | 99413 | 185000 | no |

B | 32311 | 75343 | 123000 | no |

B | 26817 | 78726 | 108000 | no |

B | 24564 | 66533 | 108000 | no |

B | 24564 | 71149 | 112900 | no |

B | 27640 | 85347 | 106000 | no |

B | 29656 | 78968 | 147500 | no |

B | 13440 | 41177 | 61000 | yes |

B | 45765 | 81227 | 320000 | yes |

B | 16680 | 72867 | 99500 | yes |

B | 17020 | 61935 | 93000 | yes |

B | 25751 | 82259 | 110000 | yes |

B | 25751 | 64568 | 100500 | yes |

- Give appropriate summary statistics (one measure of central tendency and one measure of

variation) for each of the 3 variables Appraised Land Value, Appraised Value of Improvements, and Sale Price, calculated separately for neighborhoods A and B. Important: PROVIDE ONLY ONE (APPROPRIATE) CENTRAL TENDENCY MEASURE AND ONE (APPROPRIATE) MEASURE OF VARIATION FOR EACH VARIABLE FOR EACH NEIGHBORHOOD. [6

points]

- Based on this data sample, do neighborhoods A and B differ in the number of houses with and without yards? In your answer be sure to calculate an appropriate statistic, state the result of the test, and state your (Paste the output from SPSS for the statistical test that you do in your answer, as well as stating your conclusion and writing out the appropriate statistic that supports your conclusion.) [6 points]

- Based on this data sample, do houses in neighborhoods A and B have different sale prices? (In your answer be sure to calculate an appropriate statistic, state the result of the test and state your conclusion.) (Paste the output from SPSS for the statistical test that you do in your answer, as well as stating your conclusion and writing out the appropriate statistic that supports your conclusion.) [6 points]

- Provide a correlation matrix for Appraised Land Value, Appraised Value of Improvements and Sale Price for
__neighborhood B__only (you will need to split the data to do this – in SPSS under the Data menu use the “split file” command, split by neighborhood, and select “organize output by groups”). In words,__explain__the meaning of the correlation between Sale price and Appraised Land Value__and__the meaning of the correlation between Appraised Land Value and Appraised Value of Improvements. [6 points]

__Note__: make sure you deselect “split file” after doing this question part, so that you analyzing all the cases for the next two parts.

- Imagine you are interested in the relationship between house Sale price and Appraised Land Value while controlling for any effects of Appraised Value of Improvements. Conduct a linear regression that allows you to test this relationship (using data for all the houses, i.e., from both neighborhoods). State your conclusion about the relationship, and provide the statistics that support your (Paste your SPSS output for this regression into your answer.) [6 points]

- Imagine you are interested in the relationship between house Sale price and Neighborhood, while controlling for any effects of Appraised Land Value and Appraised Value of Improvements on Sale price. Conduct a linear regression that allows you to test this relationship. State your conclusion about the relationship, and provide the statistics that support your conclusion. (Paste your SPSS output for this regression into your answer.) [6 points]

Excel Data

Neighborhood | Appraised Land Value | Appraised Value of Improvements | Sale Price | Has a yard? (yes/no) |

A | 56658 | 53806 | 255000 | no |

A | 93200 | 11121 | 422000 | no |

A | 76125 | 78172 | 290000 | no |

A | 28996 | 5864 | 305900 | no |

A | 30000 | 64831 | 118500 | yes |

A | 30000 | 50765 | 93900 | yes |

A | 46651 | 8573 | 191500 | yes |

A | 45990 | 91402 | 184000 | yes |

A | 42394 | 98181 | 168000 | yes |

A | 47751 | 3351 | 169000 | yes |

A | 63596 | 2182 | 208500 | yes |

A | 51428 | 72451 | 264000 | yes |

A | 54360 | 61934 | 237000 | yes |

A | 65376 | 34458 | 286500 | yes |

A | 42400 | 15046 | 202500 | yes |

A | 40800 | 92606 | 168000 | yes |

A | 12170 | 22786 | 375000 | yes |

A | 24637 | 90598 | 169900 | yes |

A | 30600 | 80858 | 135000 | yes |

A | 44730 | 99047 | 176000 | yes |

B | 38979 | 25946 | 140000 | no |

B | 14861 | 59258 | 74900 | no |

B | 14976 | 48957 | 57300 | no |

B | 15244 | 55169 | 87500 | no |

B | 18260 | 59267 | 82000 | no |

B | 16680 | 55525 | 78000 | no |

B | 53421 | 19792 | 175000 | no |

B | 31417 | 99413 | 185000 | no |

B | 32311 | 75343 | 123000 | no |

B | 26817 | 78726 | 108000 | no |

B | 24564 | 66533 | 108000 | no |

B | 24564 | 71149 | 112900 | no |

B | 27640 | 85347 | 106000 | no |

B | 29656 | 78968 | 147500 | no |

B | 13440 | 41177 | 61000 | yes |

B | 45765 | 81227 | 320000 | yes |

B | 16680 | 72867 | 99500 | yes |

B | 17020 | 61935 | 93000 | yes |

B | 25751 | 82259 | 110000 | yes |

B | 25751 | 64568 | 100500 | yes |

**Need Help-SPSS Project**

# Help-Probability Assignment

# Help-Probability Assignment

**QUESTION 4**

- Exercise 4.0a

You are visiting the rainforest, but unfortunately your insect repellent has run out. As a result, at each second, a mosquito lands on your neck with probability 0.5. If a mosquito lands, it will bite you with probability 0.2, and it will never bother you with probability 0.8, independently of other mosquitoes. What is the probability of being bit for the 1st time in the 5th second?

**2 points **

**QUESTION 5**

- Exercise 4.0b

You toss n = 10 fair coins, each showing heads with probability p_{1} = 0.5, independently of the other tosses. Each coin that shows tails is then tossed again, once. After following this process, what is the probability that exactly k = 6 coins show heads?

Hint: first calculate p_{2},the probability that a single coin shows heads (after following the 2-step process), and then make use of the binomial PMF.

**2 points **

**QUESTION 6**

- Exercise 5.3a

Let a continuous random variable X be given that takes values in [0, 1], and whose distribution function F satisfies

F(x) = 2x^{2} − x^{4} for 0 ≤ x ≤ 1

- Compute P (1/4 ≤ X ≤ 3/4 )

**QUESTION 10**

- Exercise 5.14

Determine the 10th percentile of a standard normal distribution

# Help-Probability Assignment

# Numerical analysis – Matlab required

**Numerical analysis – Matlab required**

Please solve the problems and show all the steps. Attach the Mathlab script as well.

# simple interest note and a simle discount note

**Simple interest note and a** **simple discount note**

What are the differences between a simple interest note and a simple discount note? Which type of note would have a higher effective rate of interest? Why?