# Linear Algebra

The University of New South Wales,

Updated On 02 Feb, 19

The University of New South Wales,

Updated On 02 Feb, 19

This course on Linear Algebra is meant for first year undergraduates or college students. It presents the subject in a visual geometric way, with special orientation to applications and understanding of key concepts. The subject naturally sits inside affine algebraic geometry. Flexibility in choosing coordinate frameworks is essential for understanding the subject. Determinants also play an important role, and these are introduced in the context of multi-vectors in the sense of Grassmann. NJ Wildberger is also the developer of Rational Trigonometry: a new and better way of learning and using trigonometry.

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4.1 ( 11 )

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 @0008 The geometry of a system of linear equations; m linear equations in n variables;

pg 2 @0139 The picture to keep in mind; The big picture;

pg 3 @0500 The kernel property; property versus set; remark on fundamental issue @0615 (see "math foundations" series);

pg 4 @1010 examples; what is a line?; what is a circle; properties instead of infinite sets;

pg 5 @1421 managing properties; statement of Properties moral;

pg 6 @1549 examples; properties of a 3-d vector;

pg 7 @1749 Subspace properties; Definition and examples;

pg 8 @2109 subspace properties of 2-d vectors;

pg 9 @2246 subspace properties of 3-d vectors;

pg 10 @2615 Definition of kernel property; definition of image property; Theorem 1; Theorem 2;

pg 11 @2754 Theorem proofs;

pg 12 @3205 subspaces in higher dimensional spaces; spanning set; equation set; hyperplane @3628 ;

pg 13 @3743 Linear transformation n-dim to m-dim; pg13_Theorem ;

pg 14 @4218 proof of pg13_Theorem;

pg 15 @4606 example (2d to 2d);

pg 16 @5148 example (3d to 2d);

pg 17 @5803 example (3d to 3d);

pg 18 @10313 example continued; remark typifies a linear transformation @010520;

pg 19 @10537 exercise 18.1;

pg 20 @10620 exercise 18.2; (THANKS to EmptySpaceEnterprise)

Sam

Sep 12, 2018

Excellent course helped me understand topic that i couldn't while attendinfg my college.

Dembe

March 29, 2019

Great course. Thank you very much.