This course on 3x3 matrices explores their applications in linear transformations of three-dimensional space, covering dilations, reflections, and rotations with various examples. The course aims to teach students how to interpret matrix/vector multiplication as linear transformations or changes of coordinates, understand the active vs. passive approach, and work with different types of transformations such as dilations, reflections, and rotations. The teaching method includes theoretical explanations, examples, and exercises to reinforce learning. This course is intended for individuals interested in deepening their understanding of linear algebra and its practical applications in geometry and physics.
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
