A bird’s eye view of linear algebra: The measure of a map — determinant | by Rohit Pandey | Nov, 2023


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Rohit Pandey

Towards Data Science

Image created with midjourney

This is the second chapter of the in-progress book on linear algebra, “A birds eye view of linear algebra”. The table of contents so far:

  1. Chapter-1: The basics
  2. Chapter-2: (Current) The measure of a map — determinants

Linear algebra is the tool of many-dimensions. No matter what you might be doing, as soon as you scale to n dimensions, linear algebra comes into the picture.

In the previous chapter, we described abstract linear maps. In this one, we roll up our sleeves and start to deal with matrices. Practical considerations like numerical stability, efficient algorithms, etc. will now start to be explored.

We discussed in the previous chapter the concept of vector spaces (basically n-dimensional collections of numbers — and more generally collections of fields) and linear maps that operate on two of those vector spaces, taking objects in one to the other.

As an example of these kinds of maps, one vector space could be the surface of the planet you’re sitting on and the other could be the surface of the table you might be sitting at. Literal maps of the world are also maps in this sense since they “map” every point on the surface of the Earth to a point on a paper or surface of a table, although they aren’t linear maps since they don’t preserve relative areas (Greenland appears much larger than it is for example in some of the projections).

An actual map of the surface of the Earth is also a map in the sense of linear algebra, but it is not a linear map. Image created with midjourney.

Once we pick a basis for the vector space (a collection of n “independent” vectors in the space; there could be infinite choices in general), all linear maps on that vector space get unique matrices assigned to them.

For the time being, let’s restrict our attention to maps that take vectors from an n-dimensional space back to the n-dimensional space (we’ll generalize later). The matrices corresponding to these linear maps are n x n (see section III of chapter 1). It might be useful to “quantify” such a linear map, express its effect on the vector…

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