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Mathematics LibreTexts

8: Orthogonality

( \newcommand{\kernel}{\mathrm{null}\,}\)

In Section [sec:5_3] we introduced the dot product in Rn and extended the basic geometric notions of length and distance. A set {f1,f2,,fm} of nonzero vectors in Rn was called an orthogonal set if fifj=0 for all ij, and it was proved that every orthogonal set is independent. In particular, it was observed that the expansion of a vector as a linear combination of orthogonal basis vectors is easy to obtain because formulas exist for the coefficients. Hence the orthogonal bases are the “nice” bases, and much of this chapter is devoted to extending results about bases to orthogonal bases. This leads to some very powerful methods and theorems. Our first task is to show that every subspace of Rn has an orthogonal basis.


This page titled 8: Orthogonality is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by W. Keith Nicholson (Lyryx Learning Inc.) via source content that was edited to the style and standards of the LibreTexts platform.

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