![]() Compare the fraction of the population that is young in the two countries both currently and, as projected by the U.S. The United States and japan are highly developed countries with quite different demographics. The three examples given above have very different properties. Demographic information like this is extraordinarily important for understanding the vitality, needs, and resources of a country. These figures were obtained from the United States Census Bureau International Data Base. The three figures below show the current population by age and gender and the projected population by age and gender in 25 years for three countries. Notice the following lines from the evaluated Mathematica notebook.ĭo you notice any connection between this rather cryptic Mathematica output and our observations above? We will return to this point later but first we look at another application. Notice in the screen shot below that when the first and second coordinates have the same absolute value but opposite signs that Ax lines up with x but is roughly one-fifth as long. Notice in the screen shot above that when the first coordinate (Joe's Pizza) of the vector x is roughly three times its second coordinate (Steve's Pizza) then Ax = x. The two screenshots below show some behavior that is worthy of note. Play with the applet a bit to see if you notice anything worthy of note. Next enter the matrix above in the Java applet. Notice that if we start with 2500 customers at each pizza restaurant in the first week then after a very few weeks Joe's Chicago Style Pizza seems to have three times as many customers each week as Steve's New York Style Pizza. As usual, we can represent this situation by a discrete dynamical systemĬlick here to open a Mathematica notebook to investigate this situation. As a result only 40% of the people who buy pizza at Steve's each week return the following week. Steve's New York Style Pizza uses lower quality cheese and doesn't have a very good sauce. Joe's Chicago Style Pizza has the better pizza and 80% of the people who buy pizza each week at Joe's return the following week. ![]() We begin our study of eigenvalues and eigenvectors by looking at two applications where eigenvalues and eigenvectors come up naturally and add significantly to our understanding.Ĭollegetown has two pizza restaurants and a large number of hungry pizza-loving students and faculty. There is a line that is mapped into zero - that is, you can determine the kernel visually. ![]() ![]() Thus, the linear transformation A is not onto and is not invertible. No matter what the value of x the value of y always lies along the same line.Notice that you can observe the following things experimentally and visually For example, change the matrix A to a singular matrix like This applet provdes a good way for students to literally see some of the ideas involved in linear equations. A dialog box will ask you for the new entry. You can change the entries in the matrix A by clicking on the entry you want to change. If you stretch the vector x then the vector y stretches as well.The vector y is always the same length as the vector x except that it has been rotated counter-clockwise by pi/2 radians.Before reading on, move the vector x to get an idea what this particular linear transformation does. As you move the vector x, the vector y = Ax also moves. By clicking or dragging on the graph you can move the vector x which is drawn on red. ![]() The elements of the matrix are displayed at the right hand side of the applet. This applet can be used to explore a linear transformation written in the form Arrange these two windows so that they overlap and you can move easily back-and-forth between them by clicking on the exposed portion of the inactive window to make it active. The number $\lambda$ is an eigenvalue of $A$ if there exists a non-zero vector $$ is called the eigenspace of $A$ corresponding to $\lambda$.Introduction to Eigenvalues and EigenvectorsĬlick here to open a new window with a Java applet. ![]()
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