4.3: Least Squares

Main Points.

It is possible that one coordinated is not proportional to another, but is very nearly so. Some multiple m will bring a very close value. The vector mx is as close to y as possible  The residual vector r is the distance btwn the tips of those two vectors. The better value of m, the smaller r will be because if mx was to equal y, r would be zero. By taking the dot product, r can be determined.

Data points often show a linear tendency put do not often lie on one single line, or are products of one another. All points can be expressed in three vectors where u contains the x coords, v the   y-intercepts, and s the y coords. The best fit line is the linear combo of u and v that gives s where u and v are multiplied by multiple m, plus r, to make up for the distance btwn that line and s. The fundamental problem of linear modeling is to find a linear combo of vectors that are as close to vector b as possible, with each vector multiplied by a multiple x, and all added to perpendicular vector r, which would ideally be zero. Then, b is in the span of the u vectors.

Conflicts.

Because a line can be drawn through any two points, doesn't that make the two points proportional to each other?

Reflection.

 Because the two vectors meet at the origin, the further they get the farther they are from each other. Therefore this concept seems very similar to local linear approximations. 

This concept seems very relevant because almost always when using data, the points don't fall on one line, making a best fit line necessary.

Main Points.

Linear Combinations is the when there is vector addition and scalar multiplication done simultaneously. When given three vectors, there are multiple ways to determine what operation must be performed on the first two to equal the third. It can be done graphically, or algebraically where x and y represent the unknown, the multiples. 

A system of equations can be points interpreted where you determine at which point the two eqns intersect. In vector interpretation, you determine which multiples are needed to produce a certain vector from two others.

Vectors can be drawn in higher dimensions, with R^3 being 3-D. Matrices are boxes of numbers, with their own special notation, with dimensions m by n. 

The span of vectors are all vectors that can be produced from those vectors via linear combos.

Linear dependency is determined by whether any vectors from the span, is a span from any vectors from that same span.

All vectors in the span of a span or list make up a subpace. The subpace's dimension is the smallest amount of vectos needed to make a subspace.

Reflection.

I don't really understand matrices but they seem to be a much more efficient way of keeping track of and adding or multiplying points then always writing out the vectors. I believe they are important for coding where numbers are used to represent letters and the given multiplier serves as the "key" for the code.

Conflicts.

I don't get linear combos because scalers and vectors are tow different things. I don't understand multiplying the matrices, the numbers seemed to be coming out of nowhere. 

10.7 modeling disease

Main Points

Modeling the spread of disease requires 3 variables S, the # of ppl who can become sick, I, the number that are sick, and R, the number of ppl who have been sick but can't become again. The model is important because it reveals the number of ppl that should be vaccinated, the amount of ppl that will be infected, and the time frame. All of this is important to monitor disease and save lives. As there are more sick, there are more "interactions" btwn I and S, which is proportional to S and I. The change in S over time is equal to the negative rate that members of S become I, which equals SI time a negative constant a. The change in the number of people sick euquals the rate ppl get sick minus the rate they are removed, because as the number of sick is dependent on S and R. The constants represent how quickly the disease spreads or how quickly ppl are removed. They must be constant because these quantities vary depending on disease or population, but are the same for each disease.

Challenges

Because the variables represent ppl from the same original population, i don't see why 3 different variables are needed. 

Reflections.

How are naturally immune people factored in?

10.6 modeling ant the interaction of two populations

Main points

To model the interaction btwn two different populations, 2 differential eqns are needed. In a predator-prey situation. the pops are both effect by the size of themself and each other.

A phrase plane is where the two varriables move with respect to a 3rd variable. The Phase trajectory is the path of a point

Challenges

If there are three variables, why can't a contour diagram be used? I don't understand how you actually graph or model the eqns.

Reflection

I think this is a very interesting topic. The idea that a graph can actually show motion, as opposed to just suggesting it (like how a straight line shows change over time), is really cool, almost 3-d. The models look like diagrams and are therefore easier to comprehend readily. 

Exponential growth and decay / Applications of ODE, equilibria, and stability

Main Points

The general solution to the differentional eqn dy/dt=ky is y=Ce^kt.  This is true becuase the deriv of e^t, and any multiple of it, is itself.  when k is positive there is growth, when negative there is decay. The graphs of these solutions are exponential curves with y intercepts that have the x axis as a horizontal asymptote. Population growth, continuously compounded interest, and other instances of growth/decay follow this model. 

An equilibrium solution holds true for all y values. The graph is therefore a horizontal line. They are found by setting deriv=0. The equilib func is stable when a small change of initail condions gives a solution that  stays close to the equilib as y gets increasingly bigger, but is unstable it veers from the equilib. Newton's law of heating and cooling state that a differently tempertured object will approach room temperature at a rate proportional to thier initial temp, therefore 2 diff temperatured objects will after a given period be very close in temp.

Challenges.

The text states that a solution is a function that is its own derivative. How is this so?

Reflection

Does Newton's law relate to the law of entropy? 

10.1 and 10.2: Differential eqns

Main Points

Differential equations are those which express information about the deriv. or rate of change of the original eqn. Instead of starting with the eqn and computing the deriv, you can use the deriv and work backwards. This method can be applied to many different type of real world problems sucha s marine harvesting, the net worth of a company, and pollution in a lake. The rate of change of a population can be expressed by the constant of proportionality times the current population times the difference btwn. the carrying capacity and the current population. The solution to a differential eqn is in fact another equation, it is the function with that deriv. The general solution is the family of functions that satisfies it, while the particular solution is the function with variables set at specific variables.

Challenges

How can an equation be the answer to another equation?

Reflection

Because we start with a deriv and work off that, isn't using differential eqns like finding the indefinite integral? 

9.6. constrained optimization

Main points

When the function has a limit, to express the "constrained" max or min you set it equal to the upper or lower level, respectively. Graphically, the limit is represented by a line that intersects with the contours of the function line. The max lies on this line because anything bellow is not reaching full potential by not hitting the limit, and everything above is disallowed because it exceeds the limit. Lagrange multipliers can also be applied to a constrained eqn to optimize it. Local max.s occur at a particular point that are greater than all immediate points and meets the requirements of the constrainment, while the global max this is true for all points of the function, vice versa for min.s. The optimization is found by plugging the function into three specific eqns. The Legrange multiplier used in these eqns is defined by the optim value of a funtion as the constrained value is increased by one unit. The Legrange multiplier is the rate of change of the increase in a function per unit. Furthermore a constrained function can be optimized by plugging it into the Legrange function and determining the critical points.

Challenges

Why do you set the limit equal to the limit instead of it having be less than or equal to? If there is a limit, like the 378,000 used in the book, how can that possibly represent a line as opposed to a point? so the legrange multiplier is not like e right, in that it is a special number? is it a special ratio? i dont understand it's significance. 

Reflections

Because the Legrange multiplier represents a rate of change, is it closely connected with the derivative function? Is it the deriv.?