Why ought to the variety of impartial row vectors exactly equal the variety of impartial column vectors for all matrices?

That is the fifth chapter of the in-progress e-book on linear algebra, “A Chook’s-Eye View of Linear Algebra”. The desk of contents thus far:

1. Chapter-1: The fundamentals
2. Chapter-2: The measure of a map — determinants
3. Chapter-3: Why is matrix multiplication the best way it’s?
4. Chapter-4: Programs of equations, linear regression and neural networks
5. Chapter-5 (present): Rank-nullity & why row rank==col rank?

We coated in earlier chapters that fashionable machine studying and AI purposes work by encoding all types of objects like photographs, textual content, audio, and many others. into excessive dimensional vector areas. And the basic software we’ve developed for working with excessive dimensional vector areas is linear algebra.

The core object in linear algebra is a linear map, that interprets one vector area to a different whereas satisfying sure properties, as described in chapter-1.

On this chapter, we’ll introduce the essential idea of the rank of a matrix/ linear map. It’s a single quantity (integer) that encodes essential details about the corresponding objects. Be aware that the determinant (chapter 2) had an identical position. And certainly, the 2 complement one another, with the rank offering additional granularity for instances the place the determinant appears to be like the identical.

Low rank approximations of noisy information units play a key position in information science purposes from recommender techniques to sign processing, so the ideas on this chapter will show helpful once we discover these areas.

All photographs by the writer.

Let’s do a fast recap of some key linear algebra ideas we’ll be utilizing within the the rest of this chapter. Most of them have been coated in chapter-1 and chapter-2.

A) Linear maps