# An Elementary Treatment of General Inner Products

by Jack E. Graver

College Mathematics Journal
January, 2011

Subject classification(s): Algebra and Number Theory | Linear Algebra | Inner Product Spaces
Applicable Course(s): 3.8 Linear/Matrix Algebra

A typical first course on linear algebra is usually restricted to vector spaces over the real numbers and the usual positive-definite inner product.  Hence, the proof that $\dim(\mathcal{S}) + \dim(\mathcal{S^\perp}) = \dim(\mathcal{V})$ is not presented in a way that generalizes to non-positive-definite inner products or to vector spaces over other fields.  In this note the author gives such a proof.

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