Advanced Linear Algebra, Lecture 5.6: Isometries

Advanced Linear Algebra, Lecture 5.6: Isometries

An isometry in a vector space is a map that preserves the norm, i.e., ||Ax-Ay||=||x-y|| for all vectors x and y. Examples include rotations, reflections, and translations. Given any isometry, we can compose it with a translation to get an isometry that fixes the zero vector. Such a map is said to be orthogonal. We show that such a map also preserves inner products, is linear, invertible, and its determinant is ±1, which can be thought of as "volume preserving". Finally, an isometry is characterized by the relation A*A=I, which in matrix form, means that the columns are an orthonormal basis.

[michelecantarelli]

Finally I understand why a matrix with orthonormal columns it's called orthogonal! Thank you for your videos!

[stansfieldmalosa]

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[elhainouni]

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