Recall that Graph 3 showed the Mandelbrot set on a real and complex plane. The transpose of A times A will always be square and symmetric, so it’s always invertible. So we have eigenvalues of 1 and 0.5 and want to find their eigenvectors to determine if one of them can satisfy Av=v. that correspond to solutions of the puzzle. Labor market statistics are very important for determining the well being of a country. This is where we perform row reduction on the matrix [R then our coordinates fall outside of the interior set. Then, this solution for c can be substituted into the equation PDnP-1u0 = un: Now, we can use our data to find c. First, we find the inverse of P: Now that we have found our c value, we can substitute it into our model PDnc = un. In the previous section we were able to create a Using the properties of inverse matrices, we find A=PDP-1. track of instead of one. Another thing to note about the above system is that it is linear. §A1.6 Solving I A]. We When x = 3, b = 2 again, so we already know the three points don’t sit on a line and our model will be an approximation at best. By setting up a matrix with the x values in the first column, y in the second, and the third as a column of 1’s (as picture below, the area of a triangle can be found by dividing the determinate by 2. Take a look at these games and read "The math behind these games". princess may be very unhappy about marrying some fellow off the street, Beware that this is not a standard MATLAB command, but it was created Project 5: Set up your own curve fitting problem and solve it. To focus on the algorithmic side instead of complicated calculations we replaced the real numbers by a simpler number system which in these games is represented by pieces on a Go board. moves with the curent, it makes it negative and it keeps its value of 1 ohm Reducing by dominance for the columns is the same EXCEPT we are now looking at it from the defense’s perspective, meaning that a lower number is better. However this example does a good job of showing the involvement of linear algebra in game theory, and a basic idea of what game theory is. In the drawing, e is just the observed vector b minus the projection p, or b – p. And the projection itself is just a combination of the columns of A — that’s why it’s in the column space after all — so it’s equal to A times some vector x-hat. It takes into consideration all unique aspects of your facial features and compares them to a database of already known faces and is then assigned a vector. we need to perform row reduction on the augmented matrix [R There are many different types of game theory problems out there, but we are going to look at a simple one that is the foundation of modern game theory. This method can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix. Source: https://www.fsb.muohio.edu/lij14/672_2015_Markov.pdf. That would certainly make the princess happy! The result of the mean will be a vector of the average of each individual component. But things don’t happen the same way when we use the third point. Files\MATLAB\R2006a\work", but double check to make sure. A two-person zero sum game is one where one person’s loss is equal to the other person’s gain. Note that since there are 2 free variables that General When someone loses their job, that means that somebody else has to pick up that labor, and soon the economy is not operating as well as it could be. For example: This equation includes both a function and its derivative . The set up looks like this: Lastly, the eigenvectors of this matrix need to be computed. can be either 0 or 1 because, as we saw In order to find which points are in the set, we put the real and complex numbers into a matrix and simplify, which allows us to find the behavior of our equations. gives us the equations I1+I2+…+In = 0 and V0 A related topic is regression analysis, which focuses more on statistical questions concerning curve fitting. solved. It seems that facial recognition software is something you would only find in a spy movie or in a top secret government building. What would the results be? another screen will come up that asks you to specify where to save the Therefore, by looking at this term, we can find the steady state population. = sfinal. The first row would be comprised of the 90% of the population that did not catch the disease and the 21% of the infected population that recovered but not did acquire immunity (thus re-joining the susceptible group). (If there notation, this is simply: sfinal = (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, This allows us to plug As we move forward, we will reach t = 2, 3, … , n, and subsequently our population will change to u2, u3, … , un. Project 2: Write a version of game Matrix Labyrinth. The goal of regression is to fit a mathematical model to a set of observed points. 1; 1; 1; 1; 1; 1; 1; 1; 1]. These This can be written as a system. If we multiply the population vector by the transformation matrix, we can find the changes in the populations as time progresses. To start, we should show you what the Mandelbrot set looks like. Once your face has been scanned via camera, scanner, etc., the image of your face represents a point in vary high dimensional plane. A very recent idea to solve this problem is the gamification of math. You will be surprised to learn that linear algebra comes in very handy in solving a puzzle you will see momentarily. We can reduce by row dominance, so we then turn to the columns. Initially, all 25 lights are turned off, so the state of the palace This in turn means finding common solutions to some “polynomial” equations of degree 1 (hyperplanes). the "Current Directory" drop down box at the top of your MATLAB window. Once computed, this vector acts like a key to a door. The factor on the left is the number of times we >> I = eye(5) If you are working from home, save the file to whatever you are using A two-person zero sum game is one where one person’s loss is equal to the other person’s gain. By finding a vector that satisfies this type, we can find a natural unemployment ratio in a given area. First, we want to find the eigenvalues, and then a vector that satisfies Av=v. The more variance a face has, then less eigenfaces are needed. Also, An is equal to PDnP-1. addition is defined as in section A1.4. This system stems from the fact that the population will start at an initial state, referred to as u0, at an initial time, t = 0. This derivation can be seen below: Since we know that An=PDnP-1, then using substitution on our original equation, Anu0 = un, we can conclude that PDnP-1u0 = un. get us from sinit to sfinal. If this lab has grabbed your attention, you may try •Thinking about a particle traveling through space, we imagine that its speed and direction of travel can be represented by a vector v in 3-dimensional Euclidean space R3. Of course, there were payoffs in the matrix for the offensive team that were greater than 5, as Play 3 was found to gain 15 yards against one type of defensive. Consider the following example of a system of linear differential equations: Right away we notice that the equation for is decoupled from the rest of the system- that is, does not depend on the functions of x or y; it only depends on itself. Similar logic would be applied to the third and fourth row or the populations that recover and become immune and the ones that become deceased respectively. If the eigenvectors returned match the ones originally set by the owner, it will unlock. From this equation, we can determine the eigenvectors (Eλ) for the eigenvalues by solving for the free variables. equation V= IR. We see Column 2 dominates Column 1, and so we eliminate Column 1 and are left with: Now, we can again reduce by dominance with rows as Row 2 dominates Row 3 and we are left with: We are left with just one row, row 2, or Play 2. Actually, the truth is that this software has become more popular than we thought. We use the equation. in MATLAB just for you to be able to do this assignment. We can see that the vector is a steady state vector because. Using this equation, you can determine the population of the cities at any given time, n, and determine whether or not the populations will stabilize. This means that the “rational choice” for the offensive coach is to pick Play 2. 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)T, We are now ready to state the puzzle in terms of

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