4.1 Dual spaces and bilinear forms

Know the basic definitions: Linear form (or functional), dual space $V^{*}$ (space of all linear forms), dual basis $\mathcal{B}^{*}$ (to a basis $\mathcal{B}$ for $V$ ); annihilator (of a subset of $V^{*}$ in $V$ ); dual (or adjoint or transpose) $T^{*}: W^{*} \rightarrow V^{*}$ of a linear transformation $T: V \rightarrow W$ (defined by $T^{*}(f)=f \circ T$ for $f \in W^{*}$ ); Bilinear form $B$ on $V$ (bilinear map $V \times V \rightarrow F$ ) left and right duality maps $B_{L}, B_{R}: V \rightarrow V^{*}$ (via $\left.B_{L}(x):=B(x, \cdot), B_{R}(x):=B(\cdot, x)\right) ;$ symmetric, alternating, skew symmetric, nondegenerate, anisotropic ( $B(x, x)=0 \Rightarrow x=0$ ) bilinear form. Orthogonal direct sum $V_{1} \perp V_{2}$, isometry $V_{1} \rightarrow V_{2}$ of spaces with bilinear forms, orthogonal $T$ (isometry $V \rightarrow V$ ). Symplectic plane for an alternate bilinear form (basis $u, v$ with $B(u, v)=1$, hence $B(u, u)=B(v, v)=0, B(v, u)=-1$ ).

Definitions for a given Bilinear Form $B:$ Matrix $\operatorname{Mat}_{\mathcal{B}}(B)$ of $B$ relative to an ordered basis $\mathcal{B}=\left\{x_{1}, \ldots, x_{n}\right\}$ of $V\left(b_{i j}=B\left(x_{i}, x_{j}\right)\right)$; left and right annihilators (or orthogonal complements) $U^{\perp, L}, U^{\perp, R}$ of a subspace $U$ (the vectors $x$ killing $U$ from the left or right, $B(x, U)=0$ or $B(U, x)=$ 0 ), left and right radical $\operatorname{Rad}_{L}(B)=\operatorname{ker}\left(B_{L}\right)=V^{\perp, L}, \operatorname{Rad}_{R}(B)=\operatorname{ker}\left(B_{R}\right)=V^{\perp, R}$, left, right orthogonality, orthosymmetric $(B(x, y)=0 \Rightarrow B(y, x)=0$; standard examples: symmetric and alternating forms). Isotropic vector $(B(x, x)=0)$.

Know Facts:

(1) $\operatorname{Bil}(V) \cong \operatorname{Hom}\left(V, V^{*}\right)$ linear map $(L$ or $R) V \rightarrow V^{*}\left(B\right.$ gives $L=B_{L}, R=B_{R} ; L$ or $R$ gives $B(x, y)=L(x)(y)$ or $B(x, y)=R(y)(x))$;

(2) IF $V$ IS FIN. DIM., nondegenerate bilinear form on $V \cong$ isomorphism ( $L$ or $R$ ) $V \rightarrow V^{*}$; a bilinear form $B$ is nondegenerate $\Leftrightarrow$ left nondegenerate (left radical zero) $\Leftrightarrow$ right nondegenerate (right radical zero) $\Leftrightarrow$ some/any matrix of $B$ is invertible ( $\operatorname{det}\left(b_{i j}\right) \neq 0$ ), $\operatorname{Mat}_{\mathcal{B}}(B)=\operatorname{Mat}_{\mathcal{B}^{*}, \mathcal{B}}\left(B_{R}\right)= \operatorname{Mat}_{\mathcal{B}^{*}, \mathcal{B}}\left(B_{L}\right)^{t}, \operatorname{Mat}_{\mathcal{B}^{\prime}}(B)=P^{t} \operatorname{Mat}_{\mathcal{B}}(B) P$ for the change-of-basis matrix $P=\operatorname{Mat}_{\mathcal{B}^{\prime}, \mathcal{B}}(I d) \Rightarrow$ the discriminant $\quad \operatorname{det}\left(\operatorname{Mat}_{\mathcal{B}}(B)\right)\left(F^{*}\right)^{2} \in F^{*} /\left(F^{*}\right)^{2} \cup\{0\}$ is a well-defined invariant of $B$; it is zero iff $B$ is degenerate.

(3) If $V$ is again finite-dimensional and $B: V \times V \rightarrow F$ a nondegenerate symmetric bilinear form, we may identify $V$ with $V^{*}$ via $B$, i.e. we may identify each $x \in V$ with $B_{L}(x)=B_{R}(x) \in V^{*}$ (since $B$ is symmetric, $B_{L}=B_{R}$ ). Making this identification, any endomorphism of $V^{*}$ can also be interpreted as an endomorphism of $V$. Then the dual $T^{*}: V \rightarrow V$ of an endomorphism $T$ of $V$ is characterized by the following property:
$B(x, T(y))=B\left(T^{*}(x), y\right)$ for all $x, y \in V$.
Basic Splitting Lemma: If $B$ is orthosymmetric, any finite-dimensional nondegenerate subspace $U$ of $V$ is an orthogonal direct summand, $V=U \perp U^{\perp}$.

Basic Structure Theorems:

(I) Any space with alternating form $B$ is a direct sum $V=P_{1} \perp \ldots \perp P_{r} \perp R$ for $R$ the radical, $P_{i}$ symplectic planes. Therefore, the rank of any matrix representing $B$ is even and its determinant is a square. In particular, there are no nondegenerate alternating forms in odd dimensions.
(II) Any vector space over a field $F$ with $\operatorname{char}(F) \neq 2$ with symmetric form is a direct sum $V=L_{1} \perp \ldots \perp L_{r} \perp R$ for $R$ the radical, $L_{i}$ anisotropic lines ( $F u_{i}$ with $B\left(u_{i}, u_{i}\right) \neq 0$ ); thus there is an orthogonal basis $\left\{u_{1}, \ldots, u_{r}, z_{r+1}, \ldots, z_{r+s}\right\}$ relative to which $B$ has diagonal matrix. Over an algebraically closed field, we can assume all $B\left(u_{i}, u_{i}\right)=1$; over the reals, we can assume all $B\left(u_{i}, u_{i}\right)= \pm 1$, with the numbers of +1 ’s, -1 ’s, and 0 ’s being invariants of $B$ by Sylvester’s Law of Inertia.

Related Problems

1. (Aug $94 \# 5$ ) If $f(x, y)$ is an alternating bilinear form on a 25 -dimensional real vector space, and $A$ is its matrix with respect to some basis, show that $\operatorname{det}(A)^{25}=0$.

2. (Aug 95 #7) Show that every element of $S O_{5}(\mathbb{R})=\left\{A \in G L_{5}(\mathbb{R}) \mid(A x, A y)=(x, y)\right.$ and $\operatorname{det}(A)= +1\}$ [where $(x, y)=x \cdot y=x^{t} y$ is the usual dot product on $\mathbb{R}^{5}$ ] has a nonzero fixed point $A x=x \neq 0$.

3. (Aug $95 \# 10$ ) It is easy to check that the space $M_{n}(\mathbb{R})$ of $n \times n$ real matrices is a real inner product space with inner product given by: $\langle A, B\rangle=\operatorname{trace}\left(A B^{t}\right)$, where $B^{t}$ denotes the transpose of the matrix $B$. Let $P$ be an invertible matrix and $T$ the linear operator on $M_{n}(\mathbb{R})$ defined by $T(A)=P^{t} A P$. Denote the adjoint of $T$ relative to this inner product by $T^{*}$. Prove that $T^{*}(A)=P A P^{t}$ for all $A$, and find necessary and sufficient conditions on the matrix $P$ that $T=T^{*}$. Justify your answer.

4. (Jan $97 \# 7$ ) Let $V$ be a finite-dimensional vector space over a field $F$, and let $\lambda$ and $\mu$ be two linear functionals on $V$. Define $B(x, y)=\lambda(x) \mu(y)-\lambda(y) \mu(x)$ for $x, y \in V$. Show that $B$ is an alternating bilinear form on $V$, and determine the possible values for its rank.

5. (Jan $00 \# 1$ ) Define the trace of an $n \times n$ real matrix, and show that the trace of the product $X Y$ of two real $n \times n$ matrices is the same as the trace of $Y X$.
   (b) Explain why the trace of such a matrix is the sum of all eigenvalues (possibly complex, with the appropriate multiplicities) of the matrix.
   (b) Explain how you would define the trace of an abstract linear operator on any finitedimensional real vector space $V$ (i.e., a linear transformation from $V$ to $V$ ).

6. (Jan $00 \# 2$ ) Let $K$ be a field and $M_{n}(K)$ the $n \times n$ matrices over $K$.
   (a) Determine all linear functionals $f: M_{n}(K) \longrightarrow K$ which are symmetric in the sense that $f(a b)=f(b a)$ for all matrices $a, b \in M_{n}(K)$.
   (b) Determine all linear functionals $f: M_{n}(K) \longrightarrow K$ which are invariant in the sense that $f\left(g a g^{-1}\right)=f(a)$ for all invertible $g \in G L_{n}(K)$.

7. (Aug $03 \# 6$ ) Let $B$ be a symmetric bilinear form on a finite-dimensional vector space $V$ over a field $F$. For a subspace $W \subseteq V$, we define the annihilator subspace of $W$ in $V$ : $W^{\perp}=\{x \in V \mid B(x, w)=0$ for every $w \in W\}$. Assume further that $B$ is nondegenerate, that is $V^{\perp}=\{0\}$. Show that:
   (a) there is a natural isomorphism of vector spaces from $V / W^{\perp}$ to the dual space $W^{*}$ of $W$;
   (b) $\operatorname{dim} W^{\perp}=\operatorname{dim} V-\operatorname{dim} W$;
   (c) $\left(W^{\perp}\right)^{\perp}=W$.

8. (Jan 19 #4) Let $F$ be a field with $\operatorname{char}(F) \neq 2$, let $V$ be a finite-dimensional vector space over $F$, and let $B$ be a symmetric bilinear form on $V$.
   (a) Prove that if $B \neq 0$, there exists $v \in V$ such that $B(v, v) \neq 0$.
   (b) Prove that for any $v \in V$ with $B(v, v) \neq 0$ there exists a subspace $W$ such that $V= F v \oplus W$ and $W \perp v$, that is, $B(w, v)=0$ for all $w \in W$.
   (c) Use (a) and (b) to prove that there is a basis $\left\{v_{n}\right\}$ of $V$ such that $B\left(v_{i}, v_{j}\right)=0$ for all $i \neq j$.