Linear Algaebra Section 4: Dot Products; Projection on a subspace; Least squares curve-fitting.

I had the most difficulty with example 4.4; I feel I may need a crash course in trigonometry! But I do appreciate the dot product is revisited later on in this class. But Least squares are very confusing, how can a single constant satisfy both equations that have no solution; is this approximation again? But if possible can we go over how to use the multiple of x and r (the residual vector) to determine the closest correct answer? also I would like to know what exactly the variables in this section represent; like what they are telling us about the vectors.

I suppose the idea of approximating solutions is better than no solution at all. As difficult as the calculations can be, they are useful. There are many situations that do not have an exact answer but approximations or the best educated guesses are the best available information we have.

Linear Algaebra Section 3: Span of a List of a Vectors

A span of a set of vectors can be linearly dependent or independent and there is the concept of the dimension of a the span of a set of vectors. What I found difficult was the idea of redundant vectors. If a combination of matrixes cancels a variable out of the final solution, than why have it in the matrix to begin with?

My concern with this section is the importance of whether a matrix is linearly dependent or independent. How does it apply to a real world situation? OR is this section suppose to show us other ways to manipulate vectors, I am having trouble realizing what the purpose of spanning matrixes is.

Linear Algebra.Section2.Linear Equations: two inerpretations

It has been so long since I have last seen vectors!!!! That is with the exception of the gradient vector. The most difficult idea was the matrix notation. I understand that an m*n matrix represents m rows and n columns, but i don't see how the Ax=b notation is logical. Otherwise Section 1 did a great job of explaining how vectors can be manipulated to show dilations, contractions, and multipliers.

I think matrixes would be useful because it allows you to solve multiple equations by calculating multiple systems simultaneously. Efficient calculations are appreciated in many places of business!

Modeling the Spread of Diseases

What is most difficult about this section are all the variables needed to follow the S-I-R model for diseases. S are those susceptible, I those infected, R those recovered/removed. Than there is a,that measures how infectious the disease is and b, is a constant that represents the rate that infected people are removed from the infected population. This gets confusing! but what does make sense is that the threshold population would be equal b/a. If S is greater than the threshold than there is an epidemic, if S is less there is no epidemic.

What is important from the S-I-R model is than it works for more than hypothetical situations. There have been historical incidents of infections diseases and it is important to know the likelyhood of if a population of people will be infected in order to control outbreaks and for public safety.

Modeling the Interaction of Two Populations

The idea behind the interactions of two populations is well explained in the robin and worms example. Essentially one population can only exist with the population of the other group, there are limits to the quantity of the groups. It is an example of sustainability and co-dependence. The hardest part though was looking at trajectories in the wr phase plane. when w,r, and t were given for dr/dt it = 1.2>0 what is the significance of this numerical value? Is this the value for the closed solution curve on the slope field?

Using math that involves two different entities is a good progression. I feel like it is bringing the "applied" portion back into calculus because the predator prey system is a real situation that exhibits a cyclic behavior. It is important to see math in more ways than just numbers and this is a connection I wanted to get out this class.

Exponential Growth and Decay. Applications and Models

All of the examples for Exponential growth and decay in 10.4 were great, but the one I had difficulty with was Continuously compounded interest. I understand that dB/dt is the rate (0.05) times the amount in the savings account, but the book says the intial amount of $1000 doesn't matter? why though, since the amount in the account would be the initial $1000. Why does the ODE not need the initial value? What is confusing is that in the solution of the ODE they do use the initial $1000 as B(sub0). Could you explain initial values and their involvement with ODE in class?

What I really enjoyed were all the applications of ODE's in section 10.5. I felt as if the book used an example from every possible situation so that every one could relate to at least one example. Similar to how economic examples will confuse me to no end, there is opportunity in drug quantity problems or physics (Newton's Law)!

I think that the only confusing situation that the class may have trouble with is the equilibrium solutions. It begins like a drug elimination ODE but the idea of stability/instability depending on if the reaction goes to infinity needs clarification. I personally have worked with biological and chemistry equilibriums and I have never come across equilibriums in math before.

By the way Chad... Amazing job on your swarming presentation, it was really interesting. I must admit though I chuckled when I saw a differential equation on the power point!

Slope Fields

The most difficult part about slope fields was really visualizing. Is a slope field really suppose to simplify things? How I see it is that at each point you have to calculate the slope of each y coordinate in the plane. It just appears to be inconvenient. What I had difficulty with was solving differential equations to find families of solution curves, I'm not sure but if there are multiple solutions, are there limits to what the solutions can be? I guess I am trying to figure why slope fields are useful and what they can show us besides the rate of change at specific points for functions; or is that the point and I am not seeing its importance?


What I did find interesting was how the idea of differential equations with initial values are incorporated into slope fields. Initial values equations represent real situations that have a real (unique) answer. Its gratifying to have math presented in reality so there is substance to the ideas we discuss.