The geometry of a system of linear equations | Wild Linear Algebra A 18 | NJ Wildberger

To a system of m equations in n variables, we can associated an m by n matrix A, and a linear transformation T from n dim space to m dim space. The kernel and rank of this transformation give us geometric insight into whether there are solutions, and if so what the solutions look like.

This video introduces subspaces of a general linear space, but does so in a rather unorthodox manner--more logically secure than the usual. Instead of talking idly about ''infinite sets'' which we have no hope of specifying, we talk rather about ''properties'', which fits more naturally with modern computer science. So while we do standard linear algebra, we approach it with a highly novel conceptual framework. This is discussed (or will be) at greater length in my MathFoundations series.

As usual, the discussion is brought down to earth by a careful look at some illustrative examples. This is a long lecture (more than an hour) so take it slowly.

CONTENT SUMMARY: pg 1: @00:08 The geometry of a system of linear equations; m linear equations in n variables;
pg 2: @01:39 The picture to keep in mind; The big picture;
pg 3: @05:00 The kernel property; property versus set; remark on fundamental issue @06:15 (see "math foundations" series);
pg 4: @10:10 examples; what is a line?; what is a circle; properties instead of infinite sets;
pg 5: @14:21 managing properties; statement of Properties moral;
pg 6: @15:49 examples; properties of a 3-d vector;
pg 7: @17:49 Subspace properties; Definition and examples;
pg 8: @21:09 subspace properties of 2-d vectors;
pg 9: @22:46 subspace properties of 3-d vectors;
pg 10: @26:15 Definition of kernel property; definition of image property; Theorem 1; Theorem 2;
pg 11: @27:54 Theorem proofs;
pg 12: @32:05 subspaces in higher dimensional spaces; spanning set; equation set; hyperplane @36:28 ;
pg 13: @37:43 Linear transformation n-dim to m-dim; pg13_Theorem ;
pg 14: @42:18 proof of pg13_Theorem;
pg 15: @46:06 example (2d to 2d);
pg 16: @51:48 example (3d to 2d);
pg 17: @58:03 example (3d to 3d);
pg 18: @1:03:13 example continued; remark: typifies a linear transformation @01:05:20;
pg 19: @1:05:37 exercise 18.1;
pg 20: @1:06:20 exercise 18.2; (THANKS to EmptySpaceEnterprise)

Video Chapters:
00:00 Introduction
5:00 The kernel property
10:10 What is a line? What is a circle;
14:20 Managing properties
15:49 Properties of vectors
17:48 Subspace properties
21:08 Subspaces of V²
22:46 Subspaces of V³
26:15 Definition of kernel property
32:05 subspaces in higher dimensional spaces
37:43 Linear transformation n-dim to m-dim
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Here are the Insights into Mathematics Playlists:

[njwildberger]

Yes, we are about to enter the second half of the WildLinAlg series (although it will be at least a few months before I start that).

[tejasnatu]

With you and Prof Tisdell, , UNSW students are really blessed..

[njwildberger]

We are told many things, some of them somewhat right. Calculus achieved its greatest height under the genius of Euler and Lagrange, and has been heading largely downhill since them. The logical weakness of a modern calculus text is hid from undergrads, but a careful examination shows errors and blemishes throughout.

As for ``the square root of two'', please see MathFoundations80 for an initial orientation.

[njwildberger]

These works that you refer to do not tackle the question of what exactly the definition of an ``infinite set'' is, except in a vague descriptive way. There is ultimately no mystery: ``infinite sets'' are an unnecessary abstraction in mathematics. All proper mathematics can be done using terms that have clear and concrete meanings, and definitions that have edges.

[njwildberger]

I wouldn't go so far as that. We might be able to construct something we call ``pi", but it isn't a part of our theory until we define it carefully. My own view on ``pi'' is hinted at in MF93: its relation to a number like 2/3 is somewhat analogous to the relation between a galaxy and a star.

[njwildberger]

The calculus as it is currently taught is, ahem, something of a blemish on the history of mathematics. Logically it is like a bucket with lots of holes in it. In my MathFoundations series I will be examining this issue in great detail, but also here in the WildLinAlg series you will get examples of how calculus is ultimately an algebraic subject, and that limits and ``real numbers'' are rather incidental to it!!

[darlove]

Professor Wildberger, infinite sets MOST CERTAINLY exist. At least according to the Zermelo-Fraenkel set theory which is the de-facto standard these days. One of them is the set of all natural numbers - this set is defined as an inductive set. One of the axioms ENSURES that at least one infinite set exists. An infinite set is a set that can be put in one-to-one correspondence with its proper subset. So N is infinite because can be put in 1-1 correspondence with all odd/even numbers.

[njwildberger]

No it isn't. If T:V^n --> V^m then a vector v has the kernel property for T means it must be in V^n, while if w has the image property for T then it means it must be in V^m. So these vectors can not in general be added.

But you are close: if v has the kernel property and T(u)=w, then T(u+v)=w; that is u and u+v have the same image w.

[aliensoup]

This lecture series have proved quite useful and enlightening, but at this present lecture I fear I am delving into "crackpot" territory when I hear brazen claims that infinite sets don't exist, and the current mathematical consensus is wrong. I am not equipped to argue one way or another, but I get nervous whenever someone starts ranting that 'everything you know is wrong' and that the rest of the world is deceiving you. I mean what comes next? The Earth is actually flat?

Ok, I get it. Computationally, infinity cannot exist, but conceptually it seems like a perfectly valid concept. I mean how difficult is it to grasp that right after n is n+1 ? If n can exist, then certainly n + 1 can, and so on. Yeah, we have Quantum Mechanics and the Plank length which would imply that the universe can be dissected only so far - it has limits. I'll keep watching, but my finger just got a little closer to the 'eject' button.

[RichardAlsenz]

You are correct in your conclusion, however it doesn't follow from your argument. I am not trying to be argumentative, just trying to restart the discussion from the assumed fundamentals.

On the issue of time and space, if we fail to define them it should not come as a surprise that the we will encounter problematic issues. The issues of memory, patience and etc. ..go play just don't expect much. Definitions are crucially important, especially when they are implied as obvious.