Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheet 39

Exercises

Exercise

Let ${}V$ be an ${}n$-dimensional real vector space, and let ${}\left\langle -,-\right\rangle$ denote a symmetric bilinear form on ${}V$ of type ${}(p,q)$. Show that

${}p+q\leq n\,$

holds.

Exercise

Give an example of a symmetric bilinear form showing that, in general, the linear subspace of maximal dimension, such that the restriction of the form is positive definite, is not uniquely determined.

Exercise

We consider on ${}\mathbb {R} ^{2}$ the symmetric bilinear form given by

${}\left\langle {\begin{pmatrix}x_{1}\\y_{1}\end{pmatrix}},{\begin{pmatrix}x_{2}\\y_{2}\end{pmatrix}}\right\rangle =x_{1}x_{2}-y_{1}y_{2}\,.$

Determine for every line ${}G$ through the origin whether the restriction of the form to the line is positive definite, negative definite, or the zero form.

Exercise

Let ${}V$ be a finite-dimensional real vector space, endowed with a symmetric bilinear form ${}\left\langle -,-\right\rangle$ of type ${}(p,q)$. Show that the negated form ${}-\left\langle -,-\right\rangle$ has the type ${}(q,p)$.

Exercise

Let ${}V$ be a finite-dimensional ${}\mathbb {R}$-vector space, endowed with a bilinear form of type ${}(p,q)$, and let ${}U\subseteq V$ be a linear subspace. Suppose that the restriction of the bilinear form to ${}U$ is of type ${}(p',q')$. Show that ${}p'\leq p$ and ${}q'\leq q$ hold.

Exercise

Let ${}V$ be an ${}n$-dimensional ${}\mathbb {R}$-vector space, endowed with a symmetric bilinear form of type ${}(p,q)$, and let ${}U\subseteq V$ denote a ${}d$-dimensional linear subspace. Suppose that the restriction of the bilinear form to ${}U$ has type ${}(p',q')$. Show

${}p'\geq d+p-n\,.$

Exercise

Give an example of a finite-dimensional real vector space ${}V$, together with a symmetric bilinear form ${}\left\langle -,-\right\rangle$ on ${}V$, and a basis ${}u_{1},\ldots ,u_{n}$ of ${}V$ such that ${}\left\langle u_{i},u_{i}\right\rangle >0$ for all ${}i=1,\ldots ,n$, but such that ${}\left\langle -,-\right\rangle$ is not positive definite

Exercise

Let ${}V$ be a finite-dimensional real vector space, endowed with a symmetric bilinear form ${}\left\langle -,-\right\rangle$ on ${}V$. Let ${}u_{1},\ldots ,u_{n}$ be an orthogonal basis of ${}V$ with the property ${}\left\langle u_{i},u_{i}\right\rangle >0$ for all ${}i=1,\ldots ,n$. Show that ${}\left\langle -,-\right\rangle$ is positive definite ist.

Exercise

Let ${}V$ be a finite-dimensional real vector space, and let ${}\left\langle -,-\right\rangle$ be a symmetric bilinear form of type ${}(p,q)$. Show that the dimension of the degeneracy space equals

$n-p-q.$

Exercise

Let ${}V$ be a finite-dimensional ${}K$-vector space, and let ${}\left\langle -,-\right\rangle$ denote a symmetric bilinear form on ${}V$. Show that the dimension of the degeneracy space is not necessarily the maximal dimension of a linear subspace with the property that the restricted form is the zero form.

Exercise

Let ${}V$ be an ${}n$-dimensional real vector space, and let ${}\left\langle -,-\right\rangle$ denote a symmetric bilinear form on ${}V$. Show that the following properties are equivalent.

1. The bilinear form is nondegenerate.
2. The Gram matrix of the bilinear form with respect to any basis is invertible.
3. The bilinear form is of type ${}(p,n-p)$ (with some ${}p\in \{1,\ldots ,n\}$).

Exercise

Determine the type of the symmetric bilinear form given by the Gram matrix

${\begin{pmatrix}3&1\\1&-5\end{pmatrix}}.$

Exercise

Determine the type of the symmetric bilinear form given by the Gram matrix

${\begin{pmatrix}2&4&1\\4&-2&3\\1&3&5\end{pmatrix}}.$

Exercise

Let ${}U$ and ${}V$ be finite-dimensional ${}\mathbb {R}$-vector spaces, endowed with symmetric bilinear forms ${}\Psi _{U}$ and ${}\Psi _{V}$.

a) Show that via

${}\Theta ((u_{1},v_{1}),(u_{2},v_{2}))=\Psi _{U}(u_{1},u_{2})+\Phi _{V}(v_{1},v_{2})\,,$

a symmetric bilinear form on ${}U\oplus V$ is given, and that ${}U$ and ${}V$ are orthogonal to each other.

b) Let ${}G$ be the Gram matrix of ${}\Psi _{U}$ with respect to a basis of ${}U$, and let ${}H$ be the Gram matrix of ${}\Psi _{V}$ with respect to a basis of ${}V$. Show that the block matrix of ${}G$ and ${}H$ is the Gram matrix of ${}\Theta$ with respect to the composed basis.

c) Suppose that the types of the bilinear forms is ${}(p,q)$ and ${}(p',q')$, respectively. Show that the type of ${}\Theta$ equals ${}(p+p',q+q')$.

Exercise

Let ${}\left\langle -,-\right\rangle$ be a symmetric bilinear form on a two-dimensional real vector space, which is described with respect to a basis by the Gram matrix

${\begin{pmatrix}0&b\\b&c\end{pmatrix}}.$

Determine the type of the form in dependence of ${}b,c$.

Exercise

Let ${}V$ be a finite-dimensional ${}K$-vector space, endowed with a bilinear form ${}\left\langle -,-\right\rangle$. Let ${}v_{1},\ldots ,v_{n}$ be a basis of ${}V$, and let ${}G$ denote the Gram matrix with respect to this basis. Let

$\Psi \colon V\longrightarrow {V}^{*},v\longmapsto {\left(w\mapsto \left\langle v,w\right\rangle \right)},$

be the corresponding linear mapping in the dual space ${}{V}^{*}$, and let ${}v_{1}^{*},\ldots ,v_{n}^{*}$ denote the dual basis of ${}{V}^{*}$. Show that the describing matrix of ${}\Psi$ with respect to these bases is the transposed matrix of ${}G$.

Exercise

Let ${}K$ be a field, and let ${}V$ and ${}W$ be finite-dimensional ${}K$-vector spaces. Show that the trace

$\operatorname {Hom} _{K}{\left(V,W\right)}\times \operatorname {Hom} _{K}{\left(W,V\right)}\longrightarrow K,(A,B)\longmapsto \operatorname {trace} {\left(A\circ B\right)},$

defines a perfect pairing, that is, ${}\operatorname {Hom} _{K}{\left(V,W\right)}$ and ${}\operatorname {Hom} _{K}{\left(W,V\right)}$ are dual to each other in a natural way.

Hand-in-exercises

Exercise (2 marks)

Let ${}V$ be an finite-dimensional ${}K$-vector space, endowed with a symmetric bilinear form ${}\left\langle -,-\right\rangle$. Let ${}M$ be the Gram matrix of the form with respect to a given basis of ${}V$. Show that the eigenspace to the eigenvalue ${}0$ of ${}M$, considered as a linear mapping of ${}V$ to ${}V$ with respect to this basis, equals the degeneracy space of the form.

Exercise (2 marks)

Determine the type of the symmetric bilinear form given by the Gram matrix

${\begin{pmatrix}6&7&-1\\7&5&6\\-1&6&-3\end{pmatrix}}.$

Exercise (3 marks)

Let ${}\left\langle -,-\right\rangle$ be a nondegenerate symmetric bilinear form of type ${}(n-q,q)$ on an ${}n$-dimensional real vector space. Let ${}v_{1},\ldots ,v_{n}$ be a basis of ${}V$, and let ${}G$ denote the Gram matrix of ${}\left\langle -,-\right\rangle$ with respect to this basis. Show that the sign of ${}\det G$ equals ${}(-1)^{q}$.

Exercise (5 marks)

Let ${}V$ be a finite-dimensional ${}\mathbb {R}$-vector space, equipped with a symmetric bilinear form of type ${}(p,q)$, and let ${}a$ denote the dimension of the degeneracy space of the form. Show that there exists a linear subspace ${}U\subseteq V$ such that the restriction of the form to ${}U$ is the zero form, and with

${}\dim _{\mathbb {R} }{\left(U\right)}={\min {\left(m,q\right)}}+a\,.$

Moreover, show that there does not exist a linear subspace of larger dimension such that the restriction is the zero form.

Exercise (3 marks)

Let ${}V$ be a finite-dimensional ${}K$-vector space with the dual space ${}{V}^{*}$. Show that the mapping

$V\times {V}^{*}\longrightarrow K,(v,f)\longmapsto f(v),$

is a perfect pairing between ${}V$ and ${}{V}^{*}$.

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