Exercise 2.6.17

Avg difficulty 3.0

Show that

\mathbf{u}\boldsymbol{\cdot}(\mathbf{v}-\operatorname{proj}_{\mathbf{u}}(\mathbf{v}))=0

and conclude every vector in $\mathbb{R}^{n}$ can be written as the sum of two vectors, one which is orthogonal and one which is parallel to the given vector.

Selinger, Matrix Theory and Linear Algebra, Vectors in Rn, The dot product, Exercise 2.6.17

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Adapted from Matrix Theory and Linear Algebra © 2017–2026 Peter Selinger, Lyryx Learning, and Ken Kuttler. Licensed under CC BY 4.0. Source: Matrix Theory and Linear Algebra. Imported at revision d74bc3a.