Main points:
• Multiple m2 x of x that reaches closest to y
○ Multiple is called the projection of y on the line spanned by x
○ Step from the tip of the projection m2 x to y is the residual vector r
○ m can only be estimated, the quality of estimation is indicated by the length of the residual vector r
• The fundamental problem of Linear Modeling
○ Least-squares curve-fitting with J parameters leads to projection onto subspaces spanned by J vectors.
§ The line y = m2 x is called the least squares fit to the data
○ The optimal linear combination is called the projection of the vector b on the subspace spanned by the vectors u1, . . ., uJ
Challenges:
I understood what the topic discussed in terms what is being solved but I didn’t understand the methods applied in the solving. I also had problems understanding the examples. In the first example I didn't understand how the m2 value (170/89) was obtained.
Reflection:
This topic area seems highly relevant because many social sciences have optimization problems. In my area of interest, economics, this section had even an example of the effect of advertising dollars on sales. This made the whole topic a lot more interesting to study, even I didn’t completely understand the examples.
Monday, November 24, 2008
Wednesday, November 19, 2008
LA 4.1 - 4.2 Vector Projection
Main points:
• Many of the equations we would like to solve, don’t have solutions
○ Therefore we look for approximate solutions
○ Finding an approximate solution is often easier in vector format
• Dot product (=scalar product)
○ The dot product of two vectors is zero if the two vectors are perpendicular (or orthogonal)
• Residual vector
Challenges:
I had a bit of troubles understanding residual vectors, mainly just the whole logic behind it. Partly it was due because I still have problems understanding what the terminology means.
Reflection:
The topics seemed ok, but I really need to spend more time with them and also be really alert on tomorrow's class. I am mainly looking forward to go through problems in tomorrow's class as I think the new things are otherwise understandable.
• Many of the equations we would like to solve, don’t have solutions
○ Therefore we look for approximate solutions
○ Finding an approximate solution is often easier in vector format
• Dot product (=scalar product)
○ The dot product of two vectors is zero if the two vectors are perpendicular (or orthogonal)
• Residual vector
Challenges:
I had a bit of troubles understanding residual vectors, mainly just the whole logic behind it. Partly it was due because I still have problems understanding what the terminology means.
Reflection:
The topics seemed ok, but I really need to spend more time with them and also be really alert on tomorrow's class. I am mainly looking forward to go through problems in tomorrow's class as I think the new things are otherwise understandable.
Monday, November 17, 2008
LA 1.3, 2.0 - 2.3, 3.0 - 3.3 Linear combinations, linear independence, span
Main points:
• Linear combination=a sum of multiples
• Linear equations: two interpretations
○ Following kind of a question: what multiplies of u and v can you add to get to w
○ Solving
§ Multiples are denoted as x and y
§ Solution is a pair (x,y) which satisfies the simultaneous system of two equations and two unknowns
• The geometry of linear equations
○ Points interpretation
○ Vector interpretation
• System of equations can have
○ No solution (two lines are parallel and not the same; and the two vectors are parallel and the vector we are trying to get to is not on that line)
○ One solution (two lines intersect in a point; and the two vectors can be combined in exactly one way to get to that vector)
○ Infinitely many solutions (two lines are the same and any point on this line works. The corresponding vectors are parallel and so is the vector we are trying to get to)
• Higher dimensions
• Matrix notation
○ Ax=b
○ An m * n matrix is a rectangular array of numbers, arranged in m rows and n columns
• Spanning
○ The span of a list of vectors is the set of all linear combinations that can be made
with those vectors
• Linear independence
○ No vector on the list is a linear combination of the other vectors on the list.
• Linear dependency
○ At least one of the vectors on the list is a linear combination of the other vectors on the list
• Subspaces and dimension
○ Subspace is a set of vectors that is the span of some list of vector
○ Dimension of a subspace is the minimum number of vectors required to span the subspace
Challenges:
I didn't understand what linear combinations are and what is their importance. I didn't understand what are the benefits and disadvantages of using vectors in contrast to system of equations. Not understanding linear combinations severely limited my understanding of spanning and rest of the topics. I am hoping to understand the things more clearly after tomorrow's class.
Reflection:
I was reminded how in math everything's builds on previously learnt things. As I didn't understand linear combinations, I had problems understanding spanning and the rest of the topics. I also came to realize how often I have had problems understanding the topics when I read them from the book, but how much easier it is for me to learn them in the classroom. For tomorrow, I'm especially interested to see how higher dimensions can be tried to "graph".
• Linear combination=a sum of multiples
• Linear equations: two interpretations
○ Following kind of a question: what multiplies of u and v can you add to get to w
○ Solving
§ Multiples are denoted as x and y
§ Solution is a pair (x,y) which satisfies the simultaneous system of two equations and two unknowns
• The geometry of linear equations
○ Points interpretation
○ Vector interpretation
• System of equations can have
○ No solution (two lines are parallel and not the same; and the two vectors are parallel and the vector we are trying to get to is not on that line)
○ One solution (two lines intersect in a point; and the two vectors can be combined in exactly one way to get to that vector)
○ Infinitely many solutions (two lines are the same and any point on this line works. The corresponding vectors are parallel and so is the vector we are trying to get to)
• Higher dimensions
• Matrix notation
○ Ax=b
○ An m * n matrix is a rectangular array of numbers, arranged in m rows and n columns
• Spanning
○ The span of a list of vectors is the set of all linear combinations that can be made
with those vectors
• Linear independence
○ No vector on the list is a linear combination of the other vectors on the list.
• Linear dependency
○ At least one of the vectors on the list is a linear combination of the other vectors on the list
• Subspaces and dimension
○ Subspace is a set of vectors that is the span of some list of vector
○ Dimension of a subspace is the minimum number of vectors required to span the subspace
Challenges:
I didn't understand what linear combinations are and what is their importance. I didn't understand what are the benefits and disadvantages of using vectors in contrast to system of equations. Not understanding linear combinations severely limited my understanding of spanning and rest of the topics. I am hoping to understand the things more clearly after tomorrow's class.
Reflection:
I was reminded how in math everything's builds on previously learnt things. As I didn't understand linear combinations, I had problems understanding spanning and the rest of the topics. I also came to realize how often I have had problems understanding the topics when I read them from the book, but how much easier it is for me to learn them in the classroom. For tomorrow, I'm especially interested to see how higher dimensions can be tried to "graph".
Monday, November 10, 2008
10.7 Modeling the spread of a disease
Main points:
• The S-I-R Model
○ S=the number of susceptibles, the people who are not yet sick but who could become sick
○ I=the number of infecteds, the people who are currently sick
○ R=the number of recovered, or removed, the people who have been sick and can no longer infect others or be reinfected
○ dS/dt=-aSI
○ dI/dt=aSI-bI
○ dR/dt=bI
○ Threshold population = b/a
§ If S0 is above --> epidemic
§ If S0 is below --> no epidemic
Challenges:
I understood the basics of the S-I-R model, but nevertheless had troubles fully understanding it. Part where trajectories in the phase plane were introduced made me confused, because I didn’t completely understand what the trajectories are. I think I will understand the topic better after tomorrow's class.
Reflection:
Even I am not doing biology, I have done it in the past and I do found it interesting to study the patterns of diseases spreading. It is going to be useful for just my general knowledge and also possibly useful as I am interested in international studies. Today's studying was spred over 2 hours which proved to be very unproductive in the end (due to several distractions) so I need to make sure that in future I allocate more specific timeframe for my math studying.
• The S-I-R Model
○ S=the number of susceptibles, the people who are not yet sick but who could become sick
○ I=the number of infecteds, the people who are currently sick
○ R=the number of recovered, or removed, the people who have been sick and can no longer infect others or be reinfected
○ dS/dt=-aSI
○ dI/dt=aSI-bI
○ dR/dt=bI
○ Threshold population = b/a
§ If S0 is above --> epidemic
§ If S0 is below --> no epidemic
Challenges:
I understood the basics of the S-I-R model, but nevertheless had troubles fully understanding it. Part where trajectories in the phase plane were introduced made me confused, because I didn’t completely understand what the trajectories are. I think I will understand the topic better after tomorrow's class.
Reflection:
Even I am not doing biology, I have done it in the past and I do found it interesting to study the patterns of diseases spreading. It is going to be useful for just my general knowledge and also possibly useful as I am interested in international studies. Today's studying was spred over 2 hours which proved to be very unproductive in the end (due to several distractions) so I need to make sure that in future I allocate more specific timeframe for my math studying.
Wednesday, November 5, 2008
10.6 Modeling the interaction of two populations
Main points:
• Predator-Prey model:
○ dw/dt=aw-cwr and dr/dt= -br+kwr
• The Phase Plane:
○ Graph of r and w against to see the growth of the populations
○ Phase plane = wr -plane on which the point moves
○ Phase trajectory = the path of the point
○ dr/dw=(-r+wr)/(w-wr)
• The slope field and equilibrium points
• Trajectories in the wr-phase plane
Challenges:
In the Robins and Worms example I didn't understand the meaning of the different constants. Later on the slope field and equilibrium points and their significance went just completely over my head. Similarly trajectories in the wr-phase plane proved to be something that I cannot grasp. Tomorrow's class is going to be difficult as I did not understand the topic at all.
Reflection:
I'm finding that for me spending an hour daily on mathematics is not enough. On the other hand I don’t think I can spend more time on mathematics, so I need to become somehow more efficient with my learning. I am thinking of getting some help at Max Center because preceptors are helpful but they don't seem to have the topics fresh in their mind so it always takes some time for them to remind themselves about the topic and then trying to explain it to me.
• Predator-Prey model:
○ dw/dt=aw-cwr and dr/dt= -br+kwr
• The Phase Plane:
○ Graph of r and w against to see the growth of the populations
○ Phase plane = wr -plane on which the point moves
○ Phase trajectory = the path of the point
○ dr/dw=(-r+wr)/(w-wr)
• The slope field and equilibrium points
• Trajectories in the wr-phase plane
Challenges:
In the Robins and Worms example I didn't understand the meaning of the different constants. Later on the slope field and equilibrium points and their significance went just completely over my head. Similarly trajectories in the wr-phase plane proved to be something that I cannot grasp. Tomorrow's class is going to be difficult as I did not understand the topic at all.
Reflection:
I'm finding that for me spending an hour daily on mathematics is not enough. On the other hand I don’t think I can spend more time on mathematics, so I need to become somehow more efficient with my learning. I am thinking of getting some help at Max Center because preceptors are helpful but they don't seem to have the topics fresh in their mind so it always takes some time for them to remind themselves about the topic and then trying to explain it to me.
Monday, November 3, 2008
10.4 & 10.5 Exponential growth and decay & Applications and modeling
Main points:
• General solution to dy/dt=ky is y=Ce^kt for any constant C
○ Exponential growth for k>0
○ Exponential decay for k<0
○ The constant C is the value of y when t is 0
• General solution to dy/dt=k(y-A) is y=A+Ce^kt for any constant C
○ Arbitrary constant C is the initial value of y-A
• An equilibrium solution
○ Constant for all values of the independent variable. The graph is a horizontal line.
○ Stable if a small change in the initial conditions gives a solution which tends toward the equilibrium as the independent variable tends to positive infinity
○ Unstable if a small change in the initial conditions gives a solution a curve which veers away from the equilibrium as the independent variable tends to positive infinity
Challenges:
I did understand the general concepts, but I still have problems applying them to real problems. Especially translating the problems to mathematics was something that I found difficult. I had problems understanding the example of Pollution in Great Lakes, so I would appreciate if we could go through it in class.
Reflection:
Today was difficult because I was really tired so I kept reading the sentences several times in order to understand them. Shows that I really need to consider whether it is worth trying to study if I am just very exhausted. I liked the examples on continuously compounded interest and company's revenue as the relate to economics.
• General solution to dy/dt=ky is y=Ce^kt for any constant C
○ Exponential growth for k>0
○ Exponential decay for k<0
○ The constant C is the value of y when t is 0
• General solution to dy/dt=k(y-A) is y=A+Ce^kt for any constant C
○ Arbitrary constant C is the initial value of y-A
• An equilibrium solution
○ Constant for all values of the independent variable. The graph is a horizontal line.
○ Stable if a small change in the initial conditions gives a solution which tends toward the equilibrium as the independent variable tends to positive infinity
○ Unstable if a small change in the initial conditions gives a solution a curve which veers away from the equilibrium as the independent variable tends to positive infinity
Challenges:
I did understand the general concepts, but I still have problems applying them to real problems. Especially translating the problems to mathematics was something that I found difficult. I had problems understanding the example of Pollution in Great Lakes, so I would appreciate if we could go through it in class.
Reflection:
Today was difficult because I was really tired so I kept reading the sentences several times in order to understand them. Shows that I really need to consider whether it is worth trying to study if I am just very exhausted. I liked the examples on continuously compounded interest and company's revenue as the relate to economics.
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