Lecture

Geometric Transformations in R2 and R3

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Description

This lecture covers geometric transformations in R2 and R3, including linear transformations, projections, and matrices. It explores the concepts of linear transformations, projections, matrix representations, and properties of projections. The lecture also delves into the trace of linear transformations and matrices, as well as specific cases such as when a transformation is a projection or when the trace is zero. Examples and proofs are provided to illustrate the theoretical concepts discussed.

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

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Linear algebra is the branch of mathematics concerning linear equations such as: linear maps such as: and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions.

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In linear algebra, linear transformations can be represented by matrices. If is a linear transformation mapping to and is a column vector with entries, then for some matrix , called the transformation matrix of . Note that has rows and columns, whereas the transformation is from to . There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors. Matrices allow arbitrary linear transformations to be displayed in a consistent format, suitable for computation.

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