Explore the applications of 3x3 matrices in this comprehensive 54-minute lecture on Linear Algebra. Delve into linear transformations of three-dimensional space, including dilations, reflections, and rotations. Learn about matrix/vector multiplication, identity transformations, and mixed dilations. Examine various examples of reflections and projections, both easy and complex. Discover rational rotations and parallel projections of vectors onto planes and lines. Investigate general formulas for reflections in planes and lines, and understand the concept of perpendicularity in this context. Gain insights into the benefits of abstraction in linear algebra and tackle practical exercises to reinforce your understanding of 3x3 matrices and their applications.
Overview
Syllabus
CONTENT SUMMARY: pg 1: @ matrix/vector multiplication; Two interpretations: linear transformation/Change of coordinates; active vs passive approach;
pg 2: @ linear transformation approach; example; columns of transformation matrix are the 3 basis vectors transformed;
pg 3: @ Identity transformation; dilations scales the entire space; dilations are a closed system under composition and addition; remark on diagonal matrices and rational numbers;
pg 4: @ mixed dilations; Mixed dilations are also a closed system under composition and addition;
pg 5: @ examples; easy reflections; reflection in a plane; reflection in a line;
pg 6: @ examples: easy projections; projection to a plane; projection to a line;
pg 7: @ examples: easy rotations;
pg 8: @ Rational rotations; half-turn formulation;
pg 9: @ parallel projection of a vector u onto a plane at arbitrary projection direction l;
pg 10: @ The parallel projection matrix; projection properties;
pg 11: @ projection example continued; projecting u onto the line l; remark that the resulting matrix is rank 1;
pg 12: @ A general reflection in a plane;
pg 13: @ A general reflection in a line;
pg 14: @ response of the general formulas in the case of perpendicular projection and reflection; introducing the notion of perpendicularity; the normal vector to a plane is read off as the coefficients of x,y,z in the cartesian formula of the plane;
pg 15: @46:26 revisit of the general formulas; the quadrance of the vector mentioned @48:20 ; remark on the benefits of abstraction @ ;
pg 16 @ exercises 11.1:2 ; THANKS to EmptySpaceEnterprise
Introduction
Identity transformation, dilations
Mixed dilations
Easy reflections
Easy projections
Easy Rotations
Rational Rotations
Projection onto plane
Projection onto line
Reflection T across line l
Perpendicular projections and reflection
Taught by
Insights into Mathematics
