= Bill Casselman
{c}
<Bill Casselman> writes research and expository mathematics connecting algebraic structure with concrete calculations. His author-hosted <Introduction to quadratic forms by Bill Casselman> explains <quadratic forms> over general <fields>, including the distinctions required in characteristic two. It develops decomposition results using <totally isotropic subspaces> and the <hyperbolic plane (quadratic form)>.<Introduction to quadratic forms by Bill Casselman>[Introduction and Proposition 2.6]{ref}
For example, over $\mathbb R$ the <quadratic form> $Q(x,y,z)=xy+z^2$ separates into a <hyperbolic plane (quadratic form)> on the first two coordinates and an <anisotropic quadratic form> on the last. The $x$-axis consists of isotropic <vectors>, but it is not invisible to the whole form: its associated <bilinear form> pairs $(1,0,0)$ nontrivially with $(0,1,0)$. Pairing such complementary directions produces the hyperbolic part of an <orthogonal direct sum>. The example helps distinguish a vector on which a <quadratic form> vanishes from a vector in the radical of its associated <bilinear form>.<Introduction to quadratic forms by Bill Casselman>[§2, hyperbolic plane, Proposition 2.6 and Corollaries 2.7–2.8]{ref}
\Image[https://personal.math.ubc.ca/~cass/cass_files/buvez-casselman-small.jpg]
{title=Bill Casselman}
{description=Portrait linked beside the author's name on https://personal.math.ubc.ca/~cass/[his University of British Columbia homepage].}
{height=300}
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