Bilinear form, as we had seen in the last chapter, is a natural generalization of the dot products of ℝ^{2} and ℝ^{3}, allowing us to introduce the important geometric concept of perpendicularity. The dot product of a vector *x* ∈ ℝ^{2} with itself is related to another important geometric concept; in fact, it is the square of the *length* of *x*. However, we do face a major difficulty if we try to introduce the idea of the length of a vector *v* as the square root of *f* (*v, v*) relative to an arbitrary bilinear form *f*. Even for symmetric bilinear forms, there may be non-zero vectors which are self-orthogonal, so that their lengths will be zero. Thus, to have a workable definition of the length of a vector available relative ...

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