LA 4.3

Main Points: When two values, x and y, that are proportional to each other but do not have equal constant values, the points are not found on the same line. However, a multiple of x can reach the closest to the y value, and this is called the projection of y on the line spanned by x. The step from x to y is caleld the residual vector, r. Because the residual vector and the quantity mx (the multiple of x) are perpendicular, the vectors equal 0. To solve for y, the dot product is necessary. The line y=mx is called the least squares fit to the data. Curve-fitting with two parameters leads to projections onto 2-dimensional subspaces. Scatter plots are used to collect data in order to find the values of m,a, and b in the equation s=ma +b. Finding the linear combination that is closest to s allows us to approximate the answer. Least-squares curve-fitting with J parameters leads to projection onto subspaces spanned by J vectors. The residual vector is perpendicular to all of the vectors in the set.

Challenges: I'm very confused about the J area of the LA. I really don't understand what J actually is or what it stands for. Also, I was confused on why the LA solved the equation in 4.3.1 the way that it did.

Reflections: This reading seemed to add onto what we were doing in class on Thursday. Even though I didn't fully understand the material, I think it will be more helpful in continuing what we were learning from the lecture on Thursday.

Reflections:

LA 4.1 and 4.2

Main Points: Some equations do not have solutions due to the number of equations being larger than the number of variables. For example, three equations with two variables will sometimes not have a solution. When this happens, an approximate solution is found. Dot products help to find approximate solutions to these equations.

Challenges: I understand the information about having to find approximate solutions, but I'm a little fuzzy as to where dot products come in to help.

Reflections: I still can't really comprehend why this is important in real life. The parts about angles and lenghts make sense, but when would you use something like this in the real world? Maybe more of what I'm asking is what kind of question/problem would lead you to use dot products to help find an approximate solution?

LA 1.3, 2.0-2.3, 3.0-3.3

Main Points: Linear combinations involve adding multiples of two vectors, or summing multiples. Problems involving this idea includes ones that ask you to find what multiples of vectors u and v can make vector w. Using simultaneous equations, a solution can be found and plotted, where the the x and y values are the intersection of the two lines on the graph. Two fundamental but different geometric interpretations can be seen when looking at the equations ax + by = e and cx + dy = f. They are: 1. what point is on the intersection of lines? and 2. What multiples are added to give vector w? This entire idea of combinations works for dimensions higher than 2 as well. Matrices are lists of vectors, and one list of vectors is in one column. An m x n vector is a rectangular array of numbers arranged in m rows and n columns. Three main iddeas accompany vectors. First, the span of a set of vectors is the set of all linear combinations that can be made with those vectors. Second, if a vector on the listt is not a linear combination of the other vectors on that list, the vector is said to have linear independence. Finally, subspace is the set of vectors that is a span of some list of vectors. The dimension of subspace is the minimum number of vectors required to span the subspace. Subspace is a very special collection of vectors.

Challenges: I'm a bit confused about the matrices. Specifically, if you have 2 sets of vectors, such as

(a b) and (e f)
(c d) (g h)

can you multiply a and g and c and f? Also, can you still multiply c and d and g and h?

Reflections: I know I've stated this in previous blog entries, but I'm not quite sure how any of this actually fits into the real world. I don't really understand exactly what a matrix is either and how it is helpful in anything that we are doing.

Section 10.7

Main Points: Differential equations can predict an outbreak of a disease epidemic as well as the level of vaccination necessary to stop the disease. The standard model used for this is the S-I-R model. S stands for number of susceptibles, or the people who are not yet sick but can become ill. I stands for the number of infecteds, or those that are currently sick. R represents the number of recovered people, or people who cannot be reinfected or continue to infect others. The number of susceptibles decreases with time as more people become infected. The sum of S, I, and R is unchanged, however, the numbers pertaining to each group can change within that certain number. Trajectories and graphs are also used in this type of problem to show how a disease is spreading. Where the rate of change of infecteds vs. susceptibles equals 0 (dI/dS = 0), a threshold population occurs. If the number of susceptibles is larger than the threshold population, an epidemic is present.

Challenges: I was a little confused wiht the constants used, and I don't really understand how the threshold population equals the proportion b/a.

Reflections: This section was pretty interesting because I always hear on the news about epidemics, but was never sure how people at the CDC and NIH decided that something was an epidemic. Now that I pretty much understand the S-I-R method, I can see how the scientists come to these conclusions.

Section 10.6

Main Points: Differential equations can be used to consider growth of two interacting populations, such as species competing for food, predator/prey, and symbiosis. This action requires a system of differential equations. This system is found by using the Lotka-Volterra equations. It states that the number of "encounters" between the two species is proportional to the product of the populations. Phase planes are used to graph the two populations against time. The point of the two populations versus eachother moves, and this path is called the phase trajectory. Slope fields are also used to see what the solution will look like. Setting the equations equal to zero helps find the equilibrium points. The shape of the trajectory tells how the populations vary with time, and a closed curve means that both populations oscillate periodically.

Challenges: I was confused on how the book came to the equations for the worm/robin example. I don't quite follow the part about the different constants either.

Reflections: The Lotka-Volterra equation is really helpful in finding how different populations depend on eachother. I'm sure that ecologists use this alot to see how different species are changing in number and how the food chain works.

10.4 and 10.5

Main Points: The general solution to to dy/dt=ky is y=Ce^kt for any constant C. For populations, the rate of grouth is usually proportional to the size of the population. In the equation dy/dt= k(y-A), k and A are constants. Also, y= A + Ce^kt. However, C is not the initial value of the variable, but the initial value of y-A. An equilibrium solution is constant for all values of the independent variable. The graph is a horizontal line. Equilibrium solutions can be identified by setting the derivative of the function to zero. It is stable if a small change in the initial conditions gives a solution which tends towards the equilibrium as the independent varibale tends to positive infinity. It is unstable if th curve veers away from the equilibrium as the independent variable tends to positive infinity. An example of equilibrium problems could involve Newton's Law of Heating and Cooling.

Challenges: The only part I had trouble understanding was the section involving dy/dt=k(y-A). I am unsure on how to use this equation.

Reflections: Differential equations can be very helpful in numerous ways, such as predicting population growth/decay, how fast a a drug can dissolve in a body, or the amount of pollutant in a lake or sky. I've already done differential equations, but I definitely need a refresher on them.