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# Linear Algebra

### Common Divisors

1. Let d; a; b; r, and q be integers. a) Suppose that d|a and d|b. Show that d|(ra + qb). b) Suppose a = qb + r. Show that the set of common divisors of a and b is the set of common divisors of b and r.

### 25 Algebra Problems : Trigonometry, Graphing, Matrices, Simplifying Expressions and Solving Equations

The flow rate V (in cm/s) of a storm drainpipe follows the following expression. Find the value of V when t = 2.500 s. a. 33.56 b. 5.42 c. 132.6 d. 22.5 A boat approached a harbor and the captain takes two bearings to calculate the distance to the harbor. Bearing 1 is read at 0 degrees and bearing 2 is read at 30 de

### Functions which do not converge uniformly on [0,1]

(See attached file for full problem description with proper equations) --- 9.3-3 Let . Use the result of exercise 4 of Section 9.1 to show that does not converge uniformly on [0,1], even though converges pointwise. ---

### Linear Algebra and Matrices: Least-Squares Solution and Normalized Error

Find the least squars solution of Ax=b, retaining five places to the right of the decimal point. Finally, verify that your least squares solution satisfies the least-squares problem and calculate the normalized error e=E/ lbl. A: ( 0 8 -1 ) ( 1 2 0 ) ( 0 0 3 ) ( 0 4 5 ) (represe

### Compact Metric and Normed Spaces

Let X be a compact metric space and Y be a normed space. Prove that if f_n belongs to C(X,Y), then lim_n f_n = f_o in the Sup norm if and only if lim_n f_n = f_o uniformly in X. [ Note: Sup norm: ||f|| = Sup||f(x)|| for every x in X.]

### Linear approximation

Use linear approximation, the tangent line approximation, to approximate the following: (56.4)^(1/3) (64.4)^(1/3) Show processes. Do not use a calculator. Note: the correct answer are different from the calculator computed values

### Three Variable Systems of Equations from Application Problems

1. The sum of three numbers is 6. The third number is the sum of the first and second numbers. The first number is one more than the thrid number. Find the numbers. 2. Sports - Alexandria High School scored 37 points in a football game. Six points are awarded for each touchdown. After each touchdown, the team can earn

### Linear algebra - Orthogonality and Projection

Q: Find the point P on the line passing through both the origin and the point 1,1,1 that is closest to the point 2,4,4. Then find the point q on the line passing through both the origin and the point 2,4,4 that is closes to the point 1,1,1

### Prove the Extended Pythagorean Theorem: E^2=lbl

We've defined the error between a vector b in Euclidean vector space R^m and its projection p onto a subspace of R^m as E=lb-pl. Prove the extended pythagorean Theorem: E^2=lbl^2 - lPl^2.

### Linear algebra: Projection - Two vectors a and b with elements

Consider two vectors a and b with elements a1,a2,a3 and b1,b2,b3 respectivley. Express the projection of b onto a in terms of these elements. Then do the same for the projection of a onto the axis of b.

### Solve the System of Equations : Matrices, Row Reductions

Solve the system of equations [ 1 1 0 1 8] [ X1] [ 2] [-1 1 2 -1 0] [ X2] [-1] [-2 0 4 6 2] [ X3] = [ 2] [0 -3 -1 1 4 ] [ 0] [X4] [3 1 2 5-1] [ 1] [X5]

### Discontinuous Counter Example

I need a counterexample for the following: If f:[a,b] -> R is ONE-TO-ONE and satisfies the intermediate value property, then f is continuous on [a,b]. I know that this is a false statement if you exclude the one-to-one property. The example I received before was f(x) = sin(1/x), but this function is not one-to-one. I am

### Mathematical modeling using systems of linear first order equations

Two tanks each hold 3 liters of salt water and are connected by two pipes (see figure below) the salt water in each tank is kept well stirred. Pure water flows into tank A at a rate of 5 liters per minute and the salt mixture exits tank B at the same rate. Salt water flows from tank A to tank B at the rate of 9 liters per min

### Loop currents in the electrical network shown in the diagram

(See attached file for full problem description with diagram) --- Electrical Networks Determine the loop currents in the electrical network shown in the diagram. ---

### Space Defined Elements

(See attached file for full problem description with symbols) --- Suppose is a matrix such that defines an element for . Show that . ---

### Discuss each of the three operations in Gaussian elimination

1. Discuss each of the three operations in Gaussian elimination. What do they have in common and how do they help solve systems of linear equations? 2. What is an Eigenvalue? What is an Eigenvector?

### Linear Algebra : Leontief Input-Output Model and Real-World Applications

1. Why did Leontief use linear algebra techniques to create his model? Can you think of alternative methods? 2. What are the main strengths of his model? 3. Does it have any limitations (that you can think of)? 4. How might the Input-Output model be useful in the real world? (In other words, would anyone except an Economist

### Mathematical System

I am having a problem drawing the table for the following system: Define a universal set U as the set of counting numbers. Form a new set that contains all possible subsets of U. This new set of subsets together with the operation of set intersection forms a mathematical system. Then I have to tell which properties that we did

### Finding the Binomial Coefficients

In the expansion (a+b)^14 find: a) The coefficient of a^10 b^4. b) The coefficient of a^6 b^8.

### System of equations and matrices

(See attached file for full problem description with equations) --- Consider the system: a) Rewrite using matrix notation b) show that the following vectors , Are solutions and they are linearly independent c)write the general solution of the system ---

### More Trouble with n-Space

How do I prove the following: Sbar - S with an over score Let S' denote the derived set and Sbar the closure of a set S in Rn. Prove that (Sbar)' = S' and Sbar is closed in Rn.

### Subsets in n-space

I've been having trouble with this and need some assistance. v - union ^ - intersection If S and T are subsets of Rn, prove that (int S) ^ (int T) = int (S^T) and (int S) v (int T) subset int(SvT)

### Using Induction to Prove an Equality

I want to use induction to prove this equality: 1 + z + z^2+...+z^n = (1 - z^(n+1))/(1 - z) for every n >= 1 How do I go about this? I started out by letting z = (a + bi), but got confused.

### Prove Exponential Identity Analyticity & Congruence Corollary

Prove that e^(z+a) = (e^z)(e^a) by applying the following Corollary: Corollary. If f and g and are analytic on a region G then f &#8801; g iff {z E G : f(z)=g(z)} has a limit point in G. Please see the attached file for the fully formatted problem.

### Gap between two consecutive squarefree numbers can be arbitrary

Problem 5. Recall that a number n is called squarefree if it is not divisible by any square > 1. Show that the gap between two consecutive squarefree numbers can be arbitrary large. (Hint: Find a positive integer m such that m is divisible by 2^2, m + 1 is divisible by 3^2, m + 2 is divisible by 5^2, m + 3 is divisible by 7^2 an

### Systems of Equations Application and Systems of Inequalities

# 41 Nickels and dimes. Windborne has 35 coins consisting of dimes and nickels. If the value of his coins is # 3.30, then how many of each type does he have? # 43 Blending fudges. The chocolate factory in Vancouver blends its double-dark-chocolate fudge, which is 35% fat, with its peanut butter fudge, which is 25% fat, to

### Functions, relations, and linear equations

1. In the real world, what might be a situation where it is preferable for the data to form a relation but not a function? There is a formula that converts temperature in degrees Celsius to temperature in degrees Fahrenheit. You are given the following data points: Fahrenheit Celsius Freezing point of water 32 0

### Show that 2006 is not the difference between two squares.

Show that 2006 is not the difference between two squares. Please see the attached file for the fully formatted problem.

### Congruence and Euler's Theorem : Use Euler's Theorem to find the last two digits of 7^1245.

Use Euler's Theorem to find the last two digits of 7^1245. Please see the attached file for the fully formatted problem.

### "The mathematical methods in the physical sciences".

Linear algebra questions from the book "The mathematical methods in the physical sciences". See attached file for full problem description.