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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.

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

A =



1 2 􀀀3 0

2 4 􀀀2 2

3 6 􀀀4 3



(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)

  1. For the following matrix



0 2 0

1 0 1

0 2 0



(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)

  1. 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)

  1. Denote by Rn the set of all n-tuples of real numbers. Rn 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 R4

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

vT (0; 1;􀀀2; 1) = 0.

(a) Prove that V is a subspace of R4.

(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)

  1. Determine the dimension of the subspace of R4 generated by the set of


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

(6 marks)

  1. The code words

u1 = 1010010; u2 = 1100001; u3 = 0101000; u4 = 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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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:
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

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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.

Value Formula: Mathematics Problems

Value Formula

Mathematics Problems
Read the following instructions in order to complete this discussion, and review the example of how to complete the math required for this assignment:

  1. Think of something you want or need for which you currently do not have the funds. It could be a vehicle, boat, horse, jewelry, property, vacation, college fund, retirement money, or something else. Pick something which cost somewhere between $2000 and $50,000.
  2. On page 270 of Elementary and Intermediate Algebra you will find the “Present Value Formula,” which computes how much money you need to start with now to achieve a desired monetary goal. Assume you will find an investment which promises somewhere between 5% and 10% interest on your money and you want to purchase your desired item in 12 years. (Remember that the higher the return, usually the riskier the investment, so think carefully before deciding on the interest rate.)
  3. State the following in your discussion:
    • The desired item
    • How much it will cost in 12 years
    • The interest rate you have chosen to go with from part 2
  4. Set up the formula and work the computational steps one by one, explaining how each step is worked, especially what the negative exponent means. Explain what the answer means.
  5. Does this formula look familiar to any other formulas you are aware of? If so, which formula(s) and how is it similar?
  6. Incorporate the following five math vocabulary words into your discussion. Use bold font to emphasize the words in your writing (Do not write definitions for the words; use them appropriately in sentences describing your math work.):
    • Power
    • Reciprocal
    • Negative exponent
    • Position
    • Rules of exponents

Your initial post should be 150-250 words in length. Respond to at least two of your classmates’ posts by Day 7. Do you agree with how they used the vocabulary? Do their answers make sense?


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