Chapter 11

Let \(U, W\) be subspaces of a metric vector space \(V\). Show that a) \(U \subseteq W \Rightarrow W^{\perp} \subseteq U^{\perp}\) b) \(U \subseteq U^{\perp \perp}\) c) \(U^{\perp}=U^{\perp \perp \perp}\)

Let \(U, W\) be subspaces of a metric vector space \(V\). Show that a) \((U+W)^{\perp}=U^{\perp} \cap W^{\perp}\) b) \((U \cap W)^{\perp}=U^{\perp}+W^{\perp}\)

Prove that the following are equivalent: a) \(V\) is nonsingular b) \(\langle u, x\rangle=\langle v, x\rangle\) for all \(x \in V\) implies \(u=v\)

Show that a metric vector space \(V\) is nonsingular if and only if the matrix \(M_{B}\) of the form is nonsingular, for every ordered basis \(\mathcal{B}\).

Let \(V\) be a finite-dimensional vector space with a bilinear form (.). We do not assume that the form is symmetric or alternate. Show that the following are cquivalent: a) \(\\{v \in V \mid\langle v, w\rangle=0\) for all \(w \in V\\}=0\) b) \(\\{v \in V \mid\langle w, v\rangle=0\) for all \(w \in V\\}=0\) Hint: Consider the singularity of the matrix of the form.

Find a diagonal matrix congruent to $$ \left[\begin{array}{ccc} 1 & 2 & 3 \\ 2 & 0 & 1 \\ 3 & 1 & -1 \end{array}\right] $$

Prove that the matrices $$ I_{2}=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \text { and } M=\left[\begin{array}{ll} 5 & 0 \\ 0 & 5 \end{array}\right] $$ are congruent over the base field \(F=Q\) of rational numbers. Find an invertible matrix \(P\) such that \(P^{t} I_{2} P=M\).

Is Minkowski space isometric to Euclidean space \(\mathbb{R}^{4}\) ?

If \(\langle,\),\(rangle is a symmetric bilinear form on V\) and char \((F) \neq 2\), show that \(Q(x)=\langle x, x\rangle / 2\) is a quadratic form.

Let \(V\) be a vector space over a field \(F\), with ordered basis \(\mathcal{B}=\left(v_{1}, \ldots, v_{n}\right)\). Let \(p\left(x_{1}, \ldots, x_{n}\right)\) be a homogeneous polynomial of degree \(d\) over \(F\), that is, a polynomial each of whose terms has degree \(d\). The \(d\)-form defined by \(p\) is the function from \(V\) to \(F\) defined as follows. If \(t=\Sigma a_{i} v_{i}\), then $$ p(v)=p\left(a_{1}, \ldots, a_{\mathrm{n}}\right) $$ (We use the same notation for the form and the polynomial.) Prove that 2forms are the same as quadratic forms.

Show that \(\mathrm{r}\) is an isometry on \(V\) if and only if \(Q(\tau v)=Q(v)\) where \(Q\) is the quadratic form associated with the bilinear form on \(V\). (Assume that \(\operatorname{char}(F) \neq 2\).)

Show that a quadratic form \(Q\) on \(V\) satisfies the parallelogram law: $$ Q(x+y)+Q(x-y)=2[Q(x)+Q(y)] $$

Show that if \(V\) is a nonsingular orthogonal geometry over a field \(F\), with char \((F) \neq 2\), then any totally isotropic subspace of \(V\) is also a totally degenerate space.

Is it true that \(V=\operatorname{rad}(V) \odot \operatorname{rad}(V)^{\perp-}\) ?

Let \(V\) be a nonsingular symplectic geometry and let \(T_{v, n}\) be a symplectic transvection. Prove that a) \(T_{\Sigma, a} T_{z, h}=T_{r, a+b}\) b) For any symplectic transformation \(\sigma\), $$ \sigma T_{v, a} \sigma^{-1}=T_{\sigma v, a} $$ c) For \(b \in F^{*}\), $$ T_{k, a}=\tau_{x, a t^{2}} $$ d) For a fixed \(v \neq 0\), the map \(a \mapsto \tau_{v, a}\) is an isomorphism from the additive group of \(F\) onto the group \(\left\\{T_{v, a} \mid a \in F\right\\} \subseteq \operatorname{Sp}(V)\).

Prove that if \(x\) is any nonsquare in a finite field \(F_{q}\), then all nonsquares have the form \(r^{2} x\), for some \(r \in F\). Hence, the product of any two nonsquares in \(F_{q}\) is a square.

Formulate Sylvester's law of inertia in terms of quadratic forms on \(V\).

Show that a two-dimensional space is a hyperbolic plane if and only if it is nonsingular and contains an isotropic vector. Assume that \(\operatorname{char}(F) \neq 2\).

a) Let \(U\) be a subspace of \(V\). Show that the inner product \(\langle x+U, y+U\rangle=\langle x, y\rangle\) on the quotient space \(V / U\) is well-defined if and only if \(U \subseteq \operatorname{rad}(V)\). b) If \(U \subseteq \operatorname{rad}(V)\), when is \(V / U\) nonsingular?

Let \(V=N \odot S\), where \(N\) is a totally degenerate space. a) Prove that \(N=\operatorname{rad}(V)\) if and only if \(S\) is nonsingular. b) If \(S\) is nonsingular, prove that \(S \approx V / \mathrm{rad}(V)\).

Let \(\operatorname{dim}(V)=\operatorname{dim}(W)\). Prove that \(V / \operatorname{rad}(V) \approx W / \operatorname{rad}(W)\) implies \(V \approx W\).

Let \(V=S \odot T\). Prove that a) \(\operatorname{rad}(V)=\operatorname{rad}(S) \odot \operatorname{rad}(T)\) b) \(V / \operatorname{rad}(V) \approx S / \operatorname{rad}(S) \odot T / \operatorname{rad}(T)\) c) \(\operatorname{dim}(\operatorname{rad}(V))=\operatorname{dim}(\operatorname{rad}(S))+\operatorname{dim}(\operatorname{rad}(T))\) d) \(V\) is nonsingular if and only if \(S\) and \(T\) are both nonsingular.

Let \(V\) be a nonsingular metric vector space. Because the Riesz representation theorem is valid in \(V\), we can define the adjoint \(\tau^{*}\) of a linear map \(\tau \in \mathcal{L}(V)\) exactly as in the case of real inner product spaces. Prove that \(T\) is an isometry if and only if it is bijective and unitary (that is, \(\mathrm{r} \mathrm{T}^{*}=\iota\) ).

If char \((F) \neq 2\), prove that \(\tau \in \mathcal{L}(V, W)\) is an isometry if and only if it is bijective and \(\langle\tau t, \tau v\rangle=\langle v, v\rangle\) for all \(v \in V\).

Let \(\mathcal{B}=\left\\{v_{1}, \ldots, v_{n}\right\\}\) be a basis for \(V\). Prove that \(r \in \mathcal{L}(V, W)\) is an isometry if and only if it is bijective and \(\left\langle\tau v_{i}, \tau v_{j}\right\rangle=\left\langle v_{i,} v_{j}\right\rangle\) for all \(i, j\).

Let \(V\) be a nonsingular orthogonal geometry and let \(\tau \in \mathcal{L}(V)\) be an isometry. a) Show that \(\operatorname{dim}(\operatorname{ker}(\iota-\tau))=\operatorname{dim}\left(\operatorname{im}(\iota-\tau)^{\perp}\right)\). b) Show that \(\operatorname{ker}(t-\tau)=\operatorname{im}(t-\tau)^{\perp}\). How would you describe \(\operatorname{ker}(t-\tau)\) in words? c) If \(\tau\) is a symmetry, what is \(\operatorname{dim}(\operatorname{ker}(t-\tau))\) ? d) Can you characterize symmetries by means of \(\operatorname{dim}(\operatorname{ker}(\iota-\tau))\) ?

Let \(V\) be a hyperbolic space of dimension \(2 \mathrm{~m}\) and let \(U\) be a hyperbolic subspace of \(V\) of dimension \(2 k\). Show that for each \(k \leq j \leq m\), there is a hyperbolie subspace \(\mathcal{H}_{2 j}\) of \(V\) for which \(U \subseteq \mathcal{H}_{2, j} \subseteq V\).

Let \(\operatorname{char}(F) \neq 2\). Prove that if \(X\) is a totally degenerate subspace of an orthogonal geometry \(V\), then \(\operatorname{dim}(X) \leq \operatorname{dim}(V) / 2\).

Prove that an orthogonal geometry \(V\) of dimension \(n\) is a hyperbolic space if and only if \(V\) is nonsingular, \(n\) is even and \(V\) contains a totally degenerate subspace of dimension \(n / 2\).