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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. A symmetric bilinear form is a bilinear form on a vector space that is symmetric. Symmetric bilinear forms are of great importance in the study of orthogonal polarity and quadrics.They are also more briefly referred to as just symmetric forms when "bilinear" is understood. They are closely related to quadratic forms; for the details of the distinction between the two, see -quadratic forms. The last two axioms only imply linearity in the first argument, but the first immediately implies linearity in the second argument then too.