# The Unapologetic Mathematician

## Bilinear Forms

April 14, 2009 - Posted by | Algebra, Linear Algebra

1. […] Inner Products Now that we’ve got bilinear forms, let’s focus in on when the base field is . We’ll also add the requirement that our […]

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2. […] Inner Products Now consider a complex vector space. We can define bilinear forms, and even ask that they be symmetric and nondegenerate. But there’s no way for such a form to […]

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3. […] start with either a bilinear or a sesquilinear form on the vector space . Let’s also pick an arbitrary basis of . I want […]

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4. […] of matrices and forms and try to tie both views of our matrix together. We’re considering a bilinear form \$ on a vector space over the real or complex numbers, which we can also think of as a linear […]

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5. […] back in on a real, finite-dimensional vector space and give it an inner product. As a symmetric bilinear form, the inner product provides us with an isomorphism . Now we can use functoriality to see what this […]

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6. […] I said above, this is a bilinear form. Further, Clairaut’s theorem tells us that it’s a symmetric form. Then the spectral […]

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7. […] this as a bilinear function which takes in two vectors and spits out a number . That is, is a bilinear form on the space of tangent vectors at […]

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8. […] the next three families of linear Lie algebras we equip our vector space with a bilinear form . We’re going to consider the endomorphisms such […]

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9. […] can now define a symmetric bilinear form on our Lie algebra by the […]

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