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Linear Algebra

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

Linear Algebra electric circuits homework

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

L2 space

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

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

Systems of Linear Algebraic Equations

(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.

Systems of Linear Equations : Row Operations and Solutions

1. Find the augmented matrix for each system of linear equations: a. 5x1 + 7x2 + 8x3 = 3 -2x1 + 4x2 + 9x3 = 3 3x1 - 6x2 + x3 = 1 b. 4x1 + x2 - 7x3 = 6 5x1 + 7x2 + 2x3 = 3 5x1 + 2x2 + 5x3 = 7 c. 3x1 - 2x2 + 2x3 = 7 5x1 + 7x2 + 3x3 = 3 -5x1 + 6x2 - 8x3 = -5 2. Using elementary row operations reduce each of the augm

Metrics and Euclidean n-Space

Consider the function defined by setting: a) show that the function defines a metric on the Euclidean n-space . Please see the attached file for the fully formatted problems.

Proof : Outer Measure

Let m'(A) = inf sum of |M_i| where i is from 1 to infinity, such that A is a subset of M_i. M_i's are disjoint. Is m'(A) = m*(A) ? m*(A) is outer measure.

Eigenvector of a linear mapping

Let T : V→V be a linear mapping and suppose that x E V is an eigenvector of T corresponding to the eigenvalue λ. Show that x is an eigenvector of T2 corresponding to the eigenvalue λ2. See attached file for full problem description and equations.