Applications of Row Reduction II - Wild Linear Algebra A

Overview

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Explore applications of row reduction in understanding linear transformations between two and three-dimensional spaces in this 57-minute lecture. Delve into the concepts of spanning sets and linearly independent vectors. Analyze examples of transformations where input and output spaces differ, using row reduction to obtain image plane equations. Examine linear transformations from 3D to 2D, focusing on mapping basis vectors and identifying vectors sent to zero. Practice determining spanning sets and linear independence through various examples and row reduction techniques. Gain insights into the fundamental concepts of linear algebra and their practical applications in vector spaces.

Syllabus

CONTENT SUMMARY: pg 1: @ 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: @ 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: @ analyse previous example using row reduction; the equation of the image plane is obtained by row reduction;
pg 4: @ example2: a linear transformation from 3d to 2d;
pg 5: @16:53 example2 continued; analysed using row reduction; importance of mapping the basis vectors @ ; some vectors are sent to zero in the transformation;
pg 6: @ example continued; row reduction;
pg 7: @ 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: @ linear independence/dependence continued; examples; a set containing the zero vector;
pg 14: @ linear dependence continued; example1;
pg 15: @ example1 continued using row reduction;
pg 16: @ linear dependence continued; example2;