9.5 Bilinear forms and tensor products

9.5 Bilinear forms and tensor products

Concept links · terms present in this machine draft; source roles are unverified: vector space · dual space

A diferent sort of application of tensor products that will turn out to be important is to the description of bilinear forms, which generalize the dual space $V^{\ast }$ of linear forms on $V$. We have:

Definition (Bilinear forms)

A bilinear form $B$ on a vector space $V$ over a field $\text{ k }\left(forus,k=\mathbf{R}or\mathbf{C}\right)$ is a map

$$B:(v,v^{\prime })\in V\times V\to B(v,v^{\prime })\in k$$

that is bilinear in both entries, $i$.e.,

$$B(v+v^{\prime },v^{\prime \prime })=B(v,v^{\prime \prime })+B(v^{\prime },v^{\prime \prime }),B(cv,v^{\prime })=cB(v,v^{\prime })$$ $$B(v,v^{\prime }+v^{\prime \prime })=B(v,v^{\prime })+B(v,v^{\prime \prime }),B(v,c{v}^{\prime })=cB(v,v^{\prime })$$

where $c\in k$

If $B(v^{\prime },v)=B(v,v^{\prime })$ the bilinear form is called symmetric, if $B(v^{\prime },v)=$ $-B(v,v^{\prime })$ it is antisymmetric.

The relation to tensor products is

Theorem 9.2

The space of bilinear forms on $V$ is isomorphic to $V^{\ast }\otimes V^{\ast }$

Proof

The map

$${\alpha }_1\otimes {\alpha }_2\in V^{\ast }\otimes V^{\ast }\to B:B(v,v^{\prime })={\alpha }_1(v){\alpha }_2(v^{\prime })$$

provides, in a basis independent way, the isomorphism we are looking for. One can show this is an isomorphism using a basis. Choosing a basis ${\mathbf{e}}_j$ of $V_{:}$ the coordinate functions $v_j={\mathbf{e}}_j^{\ast }$ provide a basis of $V^{\ast }$ , so the $v_j\otimes v_k$ will be a basis of $V^{\ast }\otimes V^{\ast }$ . The map above takes linear combinations of these to bilinear forms, and is easily seen to be one-to-one and surjective for such linear combinations. □

Given a basis ${\mathbf{e}}_j$ of $V$ and dual basis $v_j$ of $V^{\ast }$ (the coordinates), the element of $V^{\ast }\otimes V^{\ast }$ corresponding to B can be written as the sum

$$\sum _{j,k}{B}_{jk}{v}_j\otimes v_k$$

This expresses the bilinear form $B$ in terms of a matrix B with entries $B_{jk}$ which can be computed as

$$B_{jk}=B({\mathbf{e}}_j,{\mathbf{e}}_k)$$

In terms of the matrix B, the bilinear form is computed as

$$B(v,v^{\prime })=\left(\begin{array}{ccc}{v}_1& \dots & v_d\end{array}\right)\left(\begin{array}{ccc}{B}_{11}& \dots & B_{1d}\\ ⋮& ⋮& ⋮\\ B_{d1}& \dots & B_{dd}\end{array}\right)\left(\begin{array}{c}{v}_1^{\prime }\\ ⋮\\ v_d^{\prime }\end{array}\right)=\mathbf{v}\cdot \mathbf{B}{\mathbf{v}}^{\prime }$$

Previous section: 9.4 Tensor products of representations · Next section: 9.6 Symmetric and antisymmetric multilinear forms · 中文版本

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