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Linear Algebra for Data Science Using Python

Orthogonal Spaces and Complements

Explore the concepts of orthogonal spaces and complements within vector spaces and matrices. Understand how subspaces relate through orthogonality, including the orthogonal complement, row space, and null space. This lesson helps you identify orthogonal subspaces and grasp their significance in linear algebra with practical examples.

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Orthogonal subspaces

Two subspaces, UU and VV, of a vector space are orthogonal when every vector in UU is orthogonal to every vector in VV and the other way around.

Example

The xx-axis and yy-axis in R2\R^2 are orthogonal subspaces. In particular, let v=[α0]\bold v=\begin{bmatrix}\alpha\\0\end{bmatrix} and u=[0β]\bold u=\begin{bmatrix}0\\\beta\end{bmatrix} ...