Lecture Notes 1: Linear Algebra

This is the first draft of my linear algebra lecture notes. It will have to go through one or two more iterations before it becomes respectable. However I post it here, mainly for my students and for anyone else interested.

LAengin-c1

I am planning to use these notes during mu summer course. There are many differences from the usual way I handled the course before,

(1) I have tried to keep the content minimal and essential,

(2) The problems are somewhat open ended in the sense students are required to construct a part of the problem themselves which is much more fun.

A nice example is the following. One of the problem asks students to create 3 lines with a unique intersection point. There are three ways of doing it, which we discussed in the class (it is not included in the notes though)

(1) Think of the easiest scenario: A student, Yash Patel  came up with the three axis as three lines.

(2) Another simple scenario which I often use is first define the intersection point and then put the (x,y) value in LHS of any equation to get the right hand side. This was used by another student Rajan Hansora, to create a non trivial set of intersecting lines.

(3) One more method that works is to create the third line by taking the linear combination of the first two. This is interesting because of its connection with the matrix row operations.

Even more interestingly not all the three methods given above directly get translated when one wants to discuss intersection of three planes.

 

 

From Bacteria to Elephants

Mathematical rules often appear in completely unexpected places. An example of that is the graph in figure below, taken from  https://openi.nlm.nih.gov/imgs/512/85/2751747/PMC2751747_1742-4682-6-17-2.png

mass-metabolism-scale

It turns out that from unicellular organisms to huge mammals, the relationship between mass and metabolic rate are linear on the log-log scale.  Many papers have been published trying to explain this relationship till now.  But there are more interesting connections. Prof. Geoffrey West in his research on cities and corporations has found similar relationships between variety of average parameters of cities (wealth, crime rate, walking speed) against population.  Here is a link to his very interesting Ted talk.

https://www.ted.com/talks/geoffrey_west_the_surprising_math_of_cities_and_corporations.

Such a straight line fit to data is a characteristics of power laws, i.e. relationships of the form y = a xk.  Such relationships are known to exist in diverse fields, ranging from linguistics and sociology to neuroscience and geophysics. Here is a link to the Wikipedia article for the interested reader.

https://en.wikipedia.org/wiki/Power_law

In physics such power law behaviors are also known to have close connections to the theory of phase transitions and renormalization groups.