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WildLinAlg18: The geometry of a system of linear equations

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)
Length: 01:08:59

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