Applications of row reduction II | Wild Linear Algebra A 16 | NJ Wildberger

This video looks at applications of row reduction to understanding linear transformations between two and three dimensional space. It also introduces the important notions of a spanning set and a linearly independent set of vectors.

This is the16th lecture in this series on Linear Algebra by N J Wildberger.

CONTENT SUMMARY: pg 1: @00:08 More applications of row reduction; How row reduction helps us to understand some interesting aspects of linear transformations; not necessarily working with square matrices; example;
pg 2: @04:05 A transformation where the input space and the output space are not necessarily the same; example: what happens to the basis vectors under transformation?;
pg 3: @8:56 analyse previous example using row reduction; the equation of the image plane is obtained by row reduction;
pg 4: @14:55 example2: a linear transformation from 3d to 2d;
pg 5: @16:53 example2 continued; analysed using row reduction; importance of mapping the basis vectors @19:46 ; some vectors are sent to zero in the transformation;
pg 6: @23:02 example continued; row reduction;
pg 7: @26:41 Spanning sets; examples; a unique linear combination;
pg 8: @30:28 Spanning sets continued; examples; pg 9: @33:35 use of row reduction to determine whether we have a spanning set; pg 10: @39:19 Spanning sets continued; example2; not a spanning set; pg 11: @41:35 spanning sets continued; 2d space; pg 12: @43:32 linearly independent sets of vectors; examples; a linearly dependent set; pg 13: @45:42 linear independence/dependence continued; examples; a set containing the zero vector;
pg 14: @48:05 linear dependence continued; example1;
pg 15: @49:42 example1 continued using row reduction;
pg 16: @51:14 linear dependence continued; example2;

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[pickeyberry]

Video Content

00:00 Introduction
08:55 Using row reduction
14:54 Example.A linear transformation
26:41 Spanning sets
33:35 Question. How to tell if you have a spanning set
43:31 Linearly independent sets

[njwildberger]

@GANMath Linear independence occurs when there is only a trivial solution (i.e. ALL coefficients zero) to adding multiples of vectors to get zero. However in Exer. 3, there is a non-trivial solution: choose a NON-ZERO coefficient for the zero vector, and a ZERO coefficient for the other vectors.

[njwildberger]

No. In three dimensional space spanned by {e1, e2, e3}, the smaller set {e1, e2} is of course still linearly independent, but not now spanning.