Linear Algebra-2 : Vector spaces

Linear Algebra: Vector spaces


Here is the second installment of my lecture notes.


Feel free to send in suggestions questions, but as before this still needs several revisions and that this is written for mechanical and chemical engineering students.

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Lecture Notes 1: Linear Algebra

Linear Algebra Lecture Notes


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.


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:…

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From Bacteria to Elephants

Reblogged from my other blog


Mathematical rules often appear in completely unexpected places. An example of that is the graph in figure below, taken from


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.

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…

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Breaking Barriers

This year, after my very positive experience at the Olin college of how interactions outside the classroom and a more humane approach strengthens the educational process, I have been running my Linear Algebra course in quite a different manner. Considering the larger number of students and limited hours as well as based on my reading of how people learn, I have zeroed in the following. The experience is highly positive, I can see that my students are happier, more open to learning and friendlier. Here are some highlights of my course.

Every lecture we start with check-ins where 3 to 4 students volunteer to share what makes them happy or sad on the given day.  The process takes less than 10 minutes. Sometimes I also participate in sharing. This has multiple positive effects.  

  1. I get to know about the class psyche, their likes, dislikes, interests, if there is something that bothers or distracts them on that particular day, it becomes apparent.
  2. We usually get to laugh at one or the other idiosyncrasy of the class, usually it is a gripe about some test or submissions.
  3. We discuss some new information, be it a rock band, movie or sport.
  4. Students become more attentive and less distracted during this time, as it is about them.

Often, though not always, I show a short video clip (5 minutes or so), at the start of the class. I have shown, ballet, sports (cycling by Danny McAskill are amazing videos), Ted, music, Mr. Bean and similar videos so far. The advantages are,

  1. It helps improve language skills of the class.
  2. Improves general knowledge.
  3. My students are happy and relaxed and ready to start learning.

 In most lectures I try to involve as many students as possible by asking questions and inviting comments. This often takes the class in some very interesting domains. As can be seen from my lecture notes on github.

I initiate more discussions on social issues, like gender stereotypes, social taboos etc, I have found my students to be quite mature.

  Once we also played an impromptu game, of throwing a ball around to the person coming up with the best answer or comment in the class (This was a bit tricky, could have gone out of hands, but questions were reasonably challenging and the light interactions kept everyone attentive).

We laugh a lot more, sometimes on my stupidity (I once collected kids in my family and took them to watch “Ramleela” thinking that a movie with poetry by Meghani has to be nice), sometimes on remarks from the students.

What is totally weird is that my classes require much less preparations (I feel almost guilty about this) run quite spontaneously and for once we all feel that we are working towards the same goal of learning and enjoying the process.



I had an interesting experience concerning students feed-backs at Olin college that I would like to recount here. A team of students in the course “Quantitative Engineering Analysis”, QEA for short, was highly frustrated due to problems in their Mathematica code, so I sat with them to sort out the errors. The students repeatedly stated  “the problem with  Mathematica is that it does not say where exactly the error is, it just says that there is an error and then it stops working”.  Their claim appeared correct, as all my simple tests of parts of their code worked fine, but when we ran their code, the software refused to proceed and did not even inform as to where exactly the error was. The team had some Greek symbols as variables and one of the students remarked that a symbol looked different at two different places, so we decided to do away with Greek symbols as variables and chose plain English symbols instead and bingo, the code was back on track.

However their lament about the software not pointing out the location of errors started a different thought process for me. A week or two before the incident, the team leader Paul, had asked the students to give their feedback on how the course was progressing but only a few students actually gave their feedback, then Paul made an impassioned appeal to students to be more interactive so that the teaching team can make any required course correction effectively. This worked, student started voicing their concerns in constructive manner and overall the lab hours became more friendly and happier.

The Mathematica incident gave me an opportunity to re-emphasize Paul’s point. I communicated with one of the students, writing, look the way it is very frustrating to work with a software which keeps throwing errors without pointing out the source of errors, or suggestions on how those errors can be corrected, similarly  It is very frustrating to the teaching team leader if you keep lamenting that the course is not working out well for you, without pointing out what exactly is the trouble and giving suggestions on how possibly the teaching team may handle the issue.

I am reminded of this issue, as the course feedback are in progress at Ahmedabad university and I recounted the incident to my Linear Algebra class today to encourage them to give precise and constructive feedback. I believe the argument is valid for all kinds of feedback processes.

Right Words

The Ahmedabad University holds a regular event “Nalanda” where experts from outside the university engage students and faculty members in conversation. I have attended only one of them so far, the one with Mr. Anil Gupta. I have always admired him because he actually walks the talk. He walks in his Shodhyatra to the remote parts of India in harsh climates, looking for innovations and traditional knowledge in rural India.

His insight that the use of right words changes how one lives one’s life is something I couldn’t agree more with. Instead of saying “this table does not move”, say, “I am unable to move this table”. Instead of saying “My staff is not getting engaged in the goals I set”, say “I am unable to engage my staff in the goals I set”. Instead of saying “nobody gives me proper feedback”, say “I am unable to read the writing on the walls”.

What is interesting and amusing to me is that even people who claim to give very high importance to use of correct words, end up mis-reading. I don’t want to insult or cause harm but I am thoroughly amused. Thanks for the drama and entertainment.

A one sample survey

On 5th Feb. I attended a talk on maternal health in construction workers. Several  thoughts occurred to me which I note in this personal account.

This dates back to the year 2000 when I was expecting my baby, I was 31 then and according to my then doctor at the Nazareth hospital in Allahabad I needed to take extra care of myself and not continue my post-doc fellowship.  Now one of my strong opinion has been that


So I neglected my doctor’s advice and went to continue my post-doc at Regensburg, Germany. Matthias and Liz were my wonderful hosts and  equal believers in normalcy of pregnancy. I still remember that Matthias picked me up at the railway station and dropped me to my apartment and with fatherly affection watched me carrying my not so light suitcase up the stairs. Sasha, my collaborator on the other hand used to be overly caring. Some advice samples:

“ Don’t carry that heavy book, I shall carry it for you”

“Don’t sit so close to the computer, it will damage your eyes”

“Let me carry your plate back”

“Don’t drink flavored drinks” (a stranger had given me a bottle at a cafeteria as a mark of friendship, I poured it down the sink)

Apparently similar advise also went to Misha’s wife, who was also expecting. When Murthy asked how I was getting along with Sasha, my reply was “He has been mothering me”. Actually he was worse than my mother, even my mother would not have been so much concerned about me. I swum regularly, visited Paris and climbed up the Effiel  tower  and took a bus to the hospital several days after my baby was due. It used to be fun to answer in negative numbers when people asked “when are you due?”.

However the main point for this blog is about a survey I took. I returned to India in June about a month before my son was due and started visiting Nazareth again. My friend who is a nurse had advised me to go to a charity hospital, as “they shall not keep you longer than they need to”. I have found that advice quite meaningful. The doctors were very busy, often seeing 2 or 3 patients at a time. The visits were highly educative.

At the time, a researcher who was studying nutrition in pregnant women asked me if I shall fill up survey forms for her regularly. I readily agreed. What was unexpected was that the doctor I was seeing tried to guide what I should fill into the form. She suggested that I should write, I eat frequent small meals during the day. As a researcher I respect facts and so did not give-in to the doctors suggestion.

When I was listening to the seminar several thoughts came to my mind. I list them here,

  • The speaker seemed to imply that people in lower income group have less human values: In my experience people in lower income groups have more human values than middle class. For, for them interacting with others is a major part of their activity. Dominique Lapierre in the “city of joy” seems to agree with me, so does Obama in his autobiography.
  • While showing data about research it is more meaningful to show actual number of samples than percentages. 10 out of 50 is more meaningful than 20% of samples. In my example if I say 100% of women were influenced by their doctors to mis-represent  their food habits. I shall not be wrong.
  • The speaker compared the values with the best in the distribution. Sure Tamilnadu and Kerala lead in medical infrastructure and access. Comparisons should be done with the average or as a good teacher of mine suggested with median.
  • Shortcoming of the study need to be honestly accepted, research is not a race. One’s work becomes more credible if one can point out the areas where more work would help. So instead of saying, I have taken into account all possible causes, which is next to impossible one may identify the problem areas. In my mind there were two major problem areas,
  •        There are genetic factors involved, people coming from harsher climates like              deserts and mountains may have a skewed health indicators, as survival                        requires it.
  •         Demography decides nutrition and so participants have already started out                with good or bad health which had nothing to do with their work conditions.

Such factors are very difficult to incorporate, especially when the study size is mere 50 with 3-4 demography present.

  • Again in my one sample study, when a big complex was being built opposite my house one of the favorite pastime of my husband’s students who visited often and had dinner (Pratik, Ratnik, Guneshwar, Purushottam, you guys had so much fun in my absence) was to sit on the steps of our house and listen to the bhajans being sung by the construction workers. They appeared to be socially richer than the upper middle class people in our gated community complex.

I value the hard work put in by the researcher. There certainly needs to be more work in the area. I list out my concerns only because I think they may be of value. I can never thank enough peple who do good work. Thank You.