For bilinear forms, the example \(\phi_A\) given above can be viewed as prototypical.
1.
Let \(V\) be a finite-dimensional vector space, and let \(B=\basis{e}{n}\) be an ordered basis for \(V\text{.}\) Let \(\phi:V\times V\to \R\) be a bilinear form on \(V\text{.}\)
The above exercise tells us that we can study bilinear forms on a vector space by studying their matrix representations. This depends on a choice of basis, but, as one might expect, matrix representations with respect to different bases are similar.
2.
Let \(B_1,B_2\) be two ordered bases for a finite-dimensional vector space \(V\text{,}\) and let \(P=P_{B_1\leftarrow B_2}\) be the change of basis matrix for these bases. Let \(\phi:V\times V\to \R\) be a linear functional on \(V\text{.}\)
If \(A_\phi\) is the matrix of \(\phi\) with respect to the basis \(B_1\text{,}\) show that the matrix of \(\phi\) with respect to \(B_2\) is equal to \(P^TA_\phi P\text{.}\)
Bilinear forms also transform with respect to linear transformations in a manner similar to linear functionals.
3.
Let \(V\) and \(W\) be finite-dimensional vector spaces, and let \(T:V\to W\) be a linear transformation.
A bilinear form \(\phi\) on \(V\) is symmetric if \(\phi(\vv,\ww)=\phi(\ww,\vv)\) for all \(\vv,\ww\in V\text{,}\) and antisymmetric (or alternating) if \(\phi(\vv,\ww)=-\phi(\ww,\vv)\) for all \(\vv,\ww\) in \(V\text{.}\)
A bilinear form is nondegenerate if, for each nonzero vector \(\vv\in V\text{,}\) there exists a vector \(\ww\in V\) such that \(\phi(\vv,\ww)\neq 0\text{.}\) (Alternatively, for each nonzero \(\vv\in V\text{,}\) the linear functional \(\alpha(\ww)=\phi(\vv,\ww)\) is nonzero.)
A nondegenerate, antisymmetric bilinear form \(\omega\) on \(V\) is called a linear symplectic structure on \(V\text{,}\) and we call the pair \((V,\omega)\) a symplectic vector space. Symplectic structures are important in differential geometry and mathematical physics. (They can be used to encode Hamilton’s equations in classical mechanics.)
We conclude with some interesting connections between complex vector spaces and symplectic and inner product structures.
You can even check that multiplying two complex numbers is the same as multiplying the corresponding matrices, as given above!
4.
For the symplectic structure \(\omega(\vv,\ww) = v_1w_2-v_2w_1\) on \(\R^2\text{,}\) as given above, show that the matrix of \(\omega\) with respect to the standard basis is the matrix \(J_1 = \bbm 0\amp -1\\ 1\amp 0\ebm\text{.}\)
There are also interesting relationships between complex inner products, real inner products, and symplectic structures.
5.
Let \(\langle \vv,\ww\rangle\) denote the standard complex inner product on \(\C^n\text{.}\) (Recall that such an inner product is complex linear in the second argument, but for any complex scalar \(c\text{,}\) \(\langle c\vv,\ww\rangle = \overline{c}\langle \vv,\ww\rangle\text{.}\))
Write \(\vv = \mathbf{a}+i\mathbf{b}\) and \(\ww = \mathbf{c}+i\mathbf{d}\text{,}\) where \(\mathbf{a} = \langle a_1, a_2,\ldots, a_n\rangle\in\R^n\) (with similar statements for \(\mathbf{b},\mathbf{c},\mathbf{d}\)). Let \(\xx = \langle a_1,b_1,\ldots, a_n,b_n\rangle, \yy = \langle c_1,d_1,\ldots, c_n,d_n\rangle\in \R^{2n}\text{.}\)
For more reading on multilinear forms and determinants, see the 4th edition of Linear Algebra Done Right, by Sheldon Axler. For more reading on linear symplectic structures, see First Steps in Differential Geometry, by Andrew McInerney.