MA20216: Algebra 2A

Chapter 6 Bilinearity

6.1 Bilinear maps

6.1.1 Definitions and examples

Definition. Let \(U,V,W\) be vector spaces over a field \(\F \). A map \(B:U\times V\to W\) is bilinear if it is linear in each slot separately:
\(\seteqnumber{0}{6.}{0}\)
\begin{align*} B(\lambda u_1+u_2,v)&=\lambda B(u_1,v)+B(u_2,v)\\ B(u,\lambda v_1+v_2)&=\lambda B(u,v_1)+B(u,v_2), \end{align*} for all \(u,u_1,u_2\in U\), \(v,v_1,v_2\in V\) and \(\lambda \in \F \).

A bilinear map \(U\times V\to \F \) is called a bilinear pairing.

A bilinear map \(V\times V\to \F \) is called a bilinear form on \(V\).

Notation. We let \(\bil (U,V;W)\) denote the set of bilinear maps \(U\times V\to W\).

6.1.2 Bilinear forms and matrices

Definition. Let \(V\) be a vector space over \(\F \) with basis \(\cB =\lst {v}1n\) and let \(B:V\times V\to \F \) be a bilinear form. The matrix of \(B\) with respect to \(\cB \) is \(A\in M_{n\times n}(\F )\) given by
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\begin{equation*} A_{ij}=B(v_i,v_j), \end{equation*}

for \(\bw 1{i,j}n\).

Proposition 6.1. Let \(B:V\times V\to \F \) be a bilinear form with matrix \(A\) with respect to \(\cB =\lst {v}1n\). Then \(B\) is completely determined by \(A\): if \(v=\sum _{i=1}^nx_iv_i\) and \(w=\sum _{j=1}^ny_jv_j\) then
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\begin{equation*} B(v,w)=\sum _{i,j=1}^nx_iy_jA_{ij}, \end{equation*}

or, equivalently, for all \(x,y\in \F ^n\),
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\begin{equation*} B(\phi _{\cB }(x),\phi _{\cB }(y))=B_A(x,y)=\bx ^TA\by . \end{equation*}

Proposition 6.2. Let \(B:V\times V\to \F \) be a bilinear form with matrices \(A\) and \(A'\) with respect to bases \(\cB \) and \(\cB '\) of \(V\). Then
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\begin{equation*} A'=P^TAP \end{equation*}

where \(P\) is the change of basis matrix¹from \(\cB \) to \(\cB '\): thus \(\phi _P=\phi _{\cB }^{-1}\circ \phi _{\cB '}\).

¹ Algebra 1B, Definition 1.6.1.

Definition. We say that matrices \(A,B\in M_{n\times n}(\F )\) are congruent if there is \(P\in \GL (n,\F )\) such that
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\begin{equation*} B=P^TAP. \end{equation*}