Two square matrices A and B are called similar if there exists an invertible matrixV suchthatB=V−1AV.

(a) Explain in detail how similarity is related to the change of basis for- mula. What does similarity imply about the geometric relationship between B and A?

Decide, with justification, on the truth of the following propositions, both when the Universe of discourse is the set of all positive integers, and when the Universe of discourse is the set of all real numbers.

1. ∃x∀y, x < x·y

2. ∀y∃x, x < x·y

3. ∃x∀y, x = x·y

4. ∀y∃x, x = x·y

5. ∀x∃y,∃z, y2 − z2 = 4x

6. ∀x∀y∃z, z < x2 + y2

7. ∀x∃y∃z, x > yz.

A group of six friends play some games of ping-pong with these results: Amy beats Bob Bob beats Carl Frank beats Bob Amy beats Elise Carl beats Dave Elise beats Carl Elise beats Dave Frank beats Elise Frank beats Amy Consider the relation R = {hx, yi : x has beaten y}. (a) Draw the directed graph G representing R. (b) Is R reflexive? Irreflexive? Symmetric? Asymmetric? Antisymmetric? Transitive? An equivalence? An order? (c) The players want to rank themselves. Find every possible topological order of G. (d) In order to have a definitive ranking, the players want there to be only one possible topological order. Which two players should face each other? (e) The transitive closure of R (R+), is an order. Is it partial or total?

Prove or disprove: Between any n-dimensional vector space V and Rⁿ there is exactly one isomorphism T : V → Rⁿ .

dy/dx + y/x \ x³y²

Use the remaining balance method for all parts of this question.

Your company bought metal fabrication equipment that cost $350,000 using a bank loan with the following terms: 7% APR, compounded semi annually, monthly payments, with a 6 year term that pays down the equipment cost.  

Part A : After 4 years, how much would still be owing on the loan?

Part B : How much interest is paid in the 15th payment? How much principal is paid in the 15th payment?

1. (10pt) Let V and W be a vector space over R. Show that V × W together with (v0,w0)+(v1,w1)=(v0 +v1,w0 +w1) for v0,v1 ∈V, w0,w1 ∈W

   and

   λ·(v,w)=(λ·v,λ·w) for λ∈R, v∈V, w∈W is a vector space over R.

2. (5pt)LetV beavectorspaceoverR,λ,μ∈R,andu,v∈V. Provethat (λ+μ)(u+v) = ((λu+λv)+μu)+μv.

   (In your proof, carefully refer which axioms of a vector space you use for every equality. Use brackets and refer to Axiom 2 if and when you change them.)

The SIT Drug company produced two types of liquid pain killer, N(normal) and S (super). Each bottle of N requires 2 units of drug A, 1 unit of drug B and 1 unit of drug C

Each bottle of S requires 1 unit of A, 1 unit of B  and 3 units of C.The company is able to produce, each week, only 1400 units of A 800 units of B and 1800 units of C

The profit per bottle of N and S is $11 and $15 respectively.

Solve the LP model using MATLAB linprog ( ANS is S=500 and N=300 bottles) ( I just need the MATLAB codes)

prove that if you multiply 4 successive integers and take the square root of that + a number, the only number for which root(product + x) is a perfect square is 1.

Let {a_n} and {b_n} be bounded sequences. Prove that limit superior {a_n+b_n} ≦ limit superior {a_n} + limit superior{b_n}

Plot the Euler’s Method approximate solution on [0,1] for the differential equation
y* = 1 + y^2 and initial condition (a) y0 = 0 (b) y0 = 1, along with the exact solution (see
Exercise 7). Use step sizes h = 0.1 and 0.05. The exact solution is y = tan(t + c)

On the interval [-1,1], consider interpolating Runge’s function f(x) = 1/ (1 + 25x^2) By Pn(x), use computer to graph:

(c) Take 11 equally spaced nodes in [-1,1], starting at –1, ending at 1, and obtain the interpolating polynomial P10(x). Also, use 11 Chebyshev nodes in [-1,1] and obtain Pc(x), the corresponding interpolating polynomial. In the same graph, plot the three functions f(x), P10(x) and Pc(x) over the interval [-1,1] . Use different line-styles, so that f(x), P10(x) and Pc(x) look distinct.

b)Comment on the error formula for the nth degree interpolating polynomial by only computing the first three derivatives of f(x). Explain how accuracy will be lost (or gained) by interpolating f(x) at more points (by increasing n)

Let X = N^N endowed with the product topology. For x ∈ X denote x by (x₁, x₂, x₃, . . .).

(a) Decide if the function given by d : X × X → R is a metric on X where, d(x, x) = 0 and if x is not equal to y then d(x, y) = 1/n where n is the least value for which x_n is not equal to y_n. Prove your answer.

(b) Show that no compact set in X can contain a non-empty open set.

5) CHART - Create a 3-D pie chart showing the breakdown of total Hours Worked into total Overtime Hours and the NonOvertime hours located at the bottom of the database listing, row 37 columns E, F, G. The pie chart should show the values, the legend should indicate Overtime Hours and NonOvertime, and the chart title should be Total Daily Hours. Place this chart so that it will fill cells K1:R16. (Hint: When selecting the data for this chart remember that pie charts show parts of a whole so there’s no reason to include a total if you are displaying the parts making up that total.) [In spreadsheets, does it make sense to put totals at the bottom of the database?] 6) DATA TABLE - Create a Data Table that shows the average Daily Pay by gender for each department number. Start the criteria in cell E40 and the data table in cell E44. 7) SUBTOTALS - Create a Subtotal showing the total hours worked by department and the total overtime hours worked by gender within each department. Change to Outline View 3. [Which department works the most hours and do you see something odd about the overtime hours?] Copy the subtotal information to Sheet 2 using Copy/Paste Special Values (previously undisplayed/hidden data will appear when copied to Sheet 2). Remove the Subtotaling on Sheet 1 using the Remove All button before continuing. 8) GOAL SEEK - In cell E7, place a formula that will multiply the Overtime Rate (B3) by the Total Number of Overtime Hours. Perform a Goal Seek to determine the overtime rate necessary to achieve $400 in total overtime pay. Leave the goal seek in place so that it can be graded. 9) EXTRACT (Advanced Filter) - Sort the database by the employees' names in ascending order. Extract just the name and the department number (Dept. No.) of those who worked more than 8 hours. Start the criteria in cell A40 and the extract output in cell A44. 10) DATA TABLE – on SHEET 3 create a One-Variable Data Table that will take the ODD numbers from 1 to 55 and will show each of the following independent calculations: multiply by $6.64; divide by 3.245; add $8.23 to each number; subtract 4 from each number.