A new Masterclass eBook: Linear Algebra and Analysis


The Indian Academy of Sciences has published a new eBook in the Masterclass series, titled ‘Linear Algebra and Analysis ’, directed primarily at students and teachers of the undergraduate level. The eBook is a compilation of ten pedagogical articles on the topic– eight of which are sourced from Resonance, the Journal of Science Education, and two additional articles authored by Professor Rajendra Bhatia.

The articles in the eBook are written in a style that makes them appealing to readers from diverse backgrounds, approaching the subject in a manner of instruction that is different from conventional textbooks. This allows readers to correctly perceive several crucial aspects of Linear Algebra and Analysis, enabling them to encounter problems in the field with clarity and confidence.

The eBook discusses the following topics

  • Contributions of William T Gowers
  • Contributions of David Hilbert
  • How Algebraic Geometry Solves an Old Matrix Problem
  • Orthogonalisation of Vectors
  • Triangularization of a Matrix
  • Eigenvalues of AB and BA
  • The Unexpected Appearance of Pi in Diverse Problems
  • The Logarithmic Mean
  • Convolutions
  • Vibrations and Eigenvalues

About the author:

Professor Rajendra Bhatia  of the Indian Statistical Institute has made vast contributions in the fields of mathematics. Earlier, he was at the Tata Institute and in the University of Bombay. He has held visiting positions at several universities, the first of these being at the University of California, Berkeley, in 1979, and the latest being at Shanghai University in 2015.

Professor Rajendra Bhatia is a master of exposition and hence, quite inevitably, the author of several books. It is, therefore, appropriate that a collection of his essays, which have previously been published in Resonance, now be made available in a convenient format for the benefit of anyone who wishes to the comprehend the subject in an elaborate, yet simplified, manner.

Get your copy of the eBook here.

Making waves: the breaking story

Resonance is the monthly journal of science education brought out by the Indian Academy of Sciences. Most of the content is serious enough to justify the ‘Journal’ – it is meant to be a useful resource for students and teachers of science. The main target is the undergraduate group, though spill-over to interested students in the higher secondary ‘plus two’ group as well as postgraduate does happen.  The articles are refereed, and also edited for clarity and readability even after acceptance, and are expected to go beyond what is readily available in textbooks.

Right from the start, the founding fathers of Resonance felt that the serious purpose of the journal did not mean a staid appearance.  The size was chosen to fit into a hand or a small bag and carry around. (This was 1995, before everyone carried around a smartphone!) The front cover varies depending on what the issue contains, and the back cover carries the portrait of a scientist we feel that students should know more about. Authors often put in historical anecdotes, unexpected connections, digress into boxes and marginal notes. And cartoons have always been part of Resonance.  The latest issue, March 2016, is unique in the twenty year history of the journal because it brings to readers a dramatic discovery which was unannounced  (though strongly suspected) when the February issue had just come out!  The story requires a short  digression into another kind of wave.

Consider how much we all depend on electromagnetic waves –  the physicists’ name for  what most people call  radio waves or microwaves.  Maxwell gave the theory of electromagnetic waves in 1864, and Hertz produced and detected them in 1887, with our own JC Bose leaping into the fray  soon after. It took a few more decades for   radio,  television and wireless communication to appear as outstanding applications. Today, these have been eclipsed by mobile phones, the GPS, where satellites guide you to your favourite pizza place, and of course the Internet.  (The internet uses light in glass fibres, but this too is an electromagnetic wave.)

Now, there is another kind of wave, as fundamental to the physicist as the electromagnetic.  Einstein proposed his new theory of gravity in 1915, replacing Newton’s grand vision of two hundred and fifty years earlier. It took him only more one year to propose that changes in the force of gravity too must travel through space at the same speed as electromagnetic waves. The twist is that gravity is itself a property of space in Einstein’s theory!  This means that the mathematics is more complicated than what Maxwell’s waves needed. One has to understand the distortion of space moving through a space that is itself distorted.  This  complication held up people, Einstein included, for more than fifty years!  Visionaries like Wheeler in the US and Zeldovich in Russia did not wait for mathematical certainty, but plunged into the role of Einstein’s gravity for stars, and for   the universe as a whole,  from  the nineteen sixties onwards. Black holes became the symbol of this revolution.

That brings us to the announcement of February 11, 2016.   A team of hundreds of scientists and engineers who had toiled for more than a decade on LIGO stood before the world and proclaimed that the gravitational waves had finally been detected. The signature? Unimaginably minute   movements of huge mirrors in kilometre-long vacuum tubes in two separate observatories separated by most of North America, matching each other. How minute? A millionth of a millionth of a millionth of a centimetre!   The movements matched what the theorists had long said would be the death cries of two black holes merging into one. The cry is not just a figure of speech, because the frequencies are indeed in the audible range. The simulated sound would rise  in pitch and in intensity to  a shriek before lapsing into silence. This event made it to all newspapers the next day.

The Resonance team is usually in a state of tension in the middle of the month, there are articles to be copy edited, authors coaxed into responding to queries, matters of fonts and placement and all the unseen work that goes into a magazine. The gravitational wave detection forced us to abandon our plans and start afresh and in even higher gear. Fortunately, some Indian scientists have been involved in the gravitational wave scene for years, and with their help, a rather different issue was put together centred on gravitational waves, with pieces on the history, the future, the astronomy, and some of the personalities involved.  Hopefully, the brief surge of excitement which most students would have felt on reading the news will now be reinforced for some of them with a more detailed and scientific account of what this discovery means. And some of them may go on to take part in the new science which will unfold from this detection.