Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheet 38

Exercises

Exercise

Let ${}\left\langle -,-\right\rangle$ be a bilinear form on a ${}K$-vector space ${}V$. Show

${}\left\langle 0,v\right\rangle =0\,$

for all ${}v\in V$.

Exercise

Check whether the following mappings

$\mathbb {R} ^{2}\times \mathbb {R} ^{2}\longrightarrow \mathbb {R}$

are bilinear forms.

a)

${}\Psi (v,w)=\Vert {v}\Vert \,.$

b)

${}\Psi (v,w)=\Vert {v-w}\Vert \,.$

c)

${}\Psi (v,w)=\Vert {v}\Vert \cdot \Vert {w}\Vert \,.$

d)

${}\Psi (v,w)=\angle (v,w)\,.$

Exercise

Show that a real inner product is a non-degenerate bilinear form.

Exercise

Let ${}\left\langle -,-\right\rangle$ be a bilinear form on a finite-dimensional ${}K$-vector space. Show that the form is degenerate on the left if and only if it is degenerate on the right.

Exercise

We consider the linear form

$L\colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ,(x,y,z)\longmapsto x+3y-4z.$

a) Determine the vector ${}u\in \mathbb {R} ^{3}$ with the property

$\left\langle u,v\right\rangle =L(v){\text{ for all }}v\in \mathbb {R} ^{3},$

where ${}\left\langle -,-\right\rangle$ is the standard inner product.

b) Let ${}E={\left\{(x,y,z)\mid 3x-2y-5z=0\right\}}\subset \mathbb {R} ^{3}$, and let ${}\varphi =L{|}_{E}$ be the restriction of ${}L$ to ${}E$. Determine the vector ${}w\in E$ with the property

$\left\langle w,v\right\rangle =\varphi (v){\text{ for all }}v\in E,$

where ${}\left\langle -,-\right\rangle$ denotes the restriction of the standard inner products to ${}E$.

Exercise

Let ${}V$ be a Euclidean vector space, and ${}U\subseteq V$ a linear subspace, endowed with the induced inner product. Let

$f\colon V\longrightarrow \mathbb {R}$

be a linear form, and let ${}v\in V$ denote the corresponding gradient in the sense of Corollary 38.6 . Show that the gradient ${}u\in U$ of the restriction ${}f{|}_{U}$ equals the orthogonal projection of ${}v$ to ${}U$.

Exercise

Determine the Gram matrix of the standard inner product in ${}\mathbb {R} ^{2}$ with respect to the basis ${}{\begin{pmatrix}2\\-3\end{pmatrix}}$ and ${}{\begin{pmatrix}-5\\1\end{pmatrix}}$.

Exercise

Determine the Gram matrix for the determinant on ${}K^{2}$ with respect to the standard basis.

Exercise

Let ${}K$ be a field, ${}V$ a finite-dimensional ${}K$-vector space, and let ${}\left\langle -,-\right\rangle$ denote a bilinear form on ${}V$. Show that ${}\left\langle -,-\right\rangle$ is symmetric if and only if there exists a basis ${}v_{1},\ldots ,v_{n}$ of ${}V$ with

${}\left\langle v_{i},v_{j}\right\rangle =\left\langle v_{j},v_{i}\right\rangle \,$

for all ${}1\leq i,j\leq n$.

Exercise

Let ${}K$ be a field, ${}V$ a finite-dimensional ${}K$-vector space, and let ${}\left\langle -,-\right\rangle$ denote a bilinear form on ${}V$. Show that this form is symmetric if and only if its Gram matrix with respect to some basis is symmetric.

Exercise

Show that the determinant in dimension two, that is, the mapping

$K^{2}\times K^{2}\longrightarrow K,({\begin{pmatrix}x_{1}\\y_{1}\end{pmatrix}},{\begin{pmatrix}x_{2}\\y_{2}\end{pmatrix}})\longmapsto x_{1}y_{2}-x_{2}y_{1},$

is a bilinear form, which is not symmetric.

Exercise

Determine which of the following mappings ${}\varphi \colon \mathbb {R} ^{2}\times \mathbb {R} ^{2}\rightarrow \mathbb {R}$ are bilinear. In the positive case, determine whether the properties alternating and symmetric hold.

a) ${}\varphi (x,y):=x_{1}y_{1}$.

b) ${}\varphi (x,y):=x_{1}x_{2}+y_{1}y_{2}$.

c) ${}\varphi (x,y):=2x_{1}y_{2}+3x_{2}y_{1}$.

Exercise

Let ${}K$ be a field, ${}V$ a ${}K$-vector space, and let ${}\left\langle -,-\right\rangle$ denote a symmetric bilinear form on ${}V$. Show that the degeneracy space is a linear subspace of ${}V$.

Exercise

Let ${}K$ be a field, ${}V$ a finite-dimensional ${}K$-vector space, and ${}\left\langle -,-\right\rangle$ a symmetric bilinear form on ${}V$. Show that the bilinear form is not degenerated if and only if the Gram matrix of the bilinear form with respect to some basis is invertible.

Exercise

Let ${}K$ be a field, and suppose that its characteristic is not ${}2$. Let ${}\left\langle -,-\right\rangle$ denote a symmetric bilinear form on the ${}K$-vector space ${}V$. Prove that the relation

${}\left\langle v,w\right\rangle ={\frac {1}{2}}{\left(\left\langle v+w,v+w\right\rangle -\left\langle v,v\right\rangle -\left\langle w,w\right\rangle \right)}\,$

holds.

Exercise

Let ${}K$ be a field, and suppose that its characteristic is not ${}2$. Let ${}\left\langle -,-\right\rangle$ denote a symmetric bilinear form on a ${}K$-vector space ${}V$. Show

${}\left\langle v,w\right\rangle ={\frac {1}{4}}{\left(\left\langle v+w,v+w\right\rangle -\left\langle v-w,v-w\right\rangle \right)}\,.$

Exercise

Show that there might exists a bilinear form ${}\left\langle -,-\right\rangle$ on a vector space ${}V$ that it is not the zero form but

${}\left\langle v,v\right\rangle =0\,$

holds for all ${}v\in V$.

Exercise

Let ${}K$ be a field, and let ${}V$ be a finite-dimensional ${}K$-vector space endowed with a symmetric bilinear form ${}\left\langle -,-\right\rangle$. Prove that there exists an orthogonal basis on ${}V$.

Exercise

Let ${}V$ be a finite-dimensional real vector space, and let ${}\left\langle -,-\right\rangle$ denote a symmetric bilinear form on ${}V$. Show that the Gram matrix of this bilinear form with respect to a suitable basis is a diagonal matrix, whose diagonal entries are ${}1,-1$, or ${}0$.

Exercise

Let ${}M$ be a symmetric real ${}n\times n$-matrix. Show that there exists an invertible matrix ${}A$ such that

${}{A^{\text{tr}}}MA=D\,$

is a diagonal matrix, whose diagonal entries are ${}1,-1$, or ${}0$.

Exercise

Symmetric bilinear form/R/No orthogonal completion/Exercise

Exercise

Let ${}V$ be a finite-dimensional real vector space, and let ${}\left\langle -,-\right\rangle$ denote a symmetric bilinear form on ${}V$. Let ${}G$ and ${}H$ be the Gram matrices of this form with respect to the bases ${}{\mathfrak {u}}$ and ${}{\mathfrak {v}}$. Show that the determinant of ${}G$ is positive (negative, ${}0$) if and only if this holds for the determinant of ${}H$.

Exercise

a) Show that the sum of two bilinear forms ${}\Psi _{1}$ and ${}\Psi _{2}$ on a ${}K$-vector space ${}V$ is again a bilinear form.

b) Show that a scalar multiple of a bilinear form is again a bilinear form.

Exercise

Show that the set of all bilinear forms on a ${}K$-vector space ${}V$ form a ${}K$-vector space.

Exercise

Let ${}V$ be a finite-dimensional ${}K$-vector space. Show that there exists a natural isomorphy

$\operatorname {Hom} _{K}{\left(V,{V}^{*}\right)}\longrightarrow \operatorname {Bilin} _{}{\left(V\right)}.$

As in the case of an inner product, we call s linear mapping that respects a given bilinear form, an isometry.

Let ${}V$ and ${}W$ be vector spaces over ${}K$, both endowed with a bilinear form ${}\Phi _{V}$ and ${}\Phi _{W}$, A ${}K$-linear mapping ${}f\colon V\rightarrow W$ is called an isometry if

${}\Phi _{W}(f(u),f(v))=\Phi _{V}(u,v)\,$

holds for all

${}u,v\in V$.

Exercise

Let ${}U,V$ be vector spaces over ${}K$, equipped with bilinear forms ${}\Phi _{U}$ and ${}\Phi _{V}$, and let ${}\varphi \colon U\rightarrow V$ denote an isometry. Is ${}\varphi$ always injective?

Exercise

Let ${}U,V,W$ be vector spaces over ${}K$, each equipped with a bilinear form ${}\Phi _{U},\Phi _{V},\Phi _{W}$. Show that the following statements hold.

a) The identity ${}V\rightarrow V$ is an isometry.

b) If ${}\varphi \colon U\rightarrow V$ is a bijective isometry, then its inverse mapping ${}\varphi ^{-1}$ is an isometry.

c) If ${}\varphi \colon U\rightarrow V$ and ${}\psi \colon V\rightarrow W$ are isometries, then also the composition ${}\psi \circ \varphi$ is an isometry.

Exercise

Let ${}K$ be a field, and suppose that its characteristic is not ${}2$. Let ${}\left\langle -,-\right\rangle$ be a bilinear form on a ${}K$-vector space ${}V$, which is symmetric as well as alternating. Show that it is the zero-form.

Exercise

Let ${}V$ be a ${}K$-vector space, equipped with a bilinear form ${}\Phi$. Show that the set of all isometries on ${}V$ forms a group under the composition of mappings.

Exercise

Let ${}V$ be a ${}\mathbb {C}$-vector space, and let

$\varphi \colon V\longrightarrow V$

and

$\psi \colon V\longrightarrow V$

denote antilinear mappings. Show that the composition ${}\varphi \circ \psi$ is linear.

Exercise

Show that for an hermitian form ${}\left\langle -,-\right\rangle$ on a ${}\mathbb {C}$-vector space ${}V$, the values ${}\left\langle v,v\right\rangle$ for ${}v\in V$ are always real.

Exercise

Show that a sesquilinear form ${}\left\langle -,-\right\rangle$ on a ${}\mathbb {C}$-vector space ${}V$ is hermitian if and only if the Gram matrix of the form with respect to some basis of ${}V$ is hermitian.

Hand-in-exercises

Exercise (4 marks)

Determine which of the following mappings ${}\varphi \colon \mathbb {R} ^{2}\times \mathbb {R} ^{2}\rightarrow \mathbb {R}$ are bilinear. In the positive case, determine whether the properties alternating and symmetric hold.

a) ${}\varphi (x,y):=x_{1}-y_{1}$.

b) ${}\varphi (x,y):=x_{1}y_{1}-x_{2}y_{2}$.

c) ${}\varphi (x,y):=2x_{1}y_{2}-2x_{2}y_{1}$.

Exercise (3 marks)

Determine the Gram matrix of the standard inner product in ${}\mathbb {R} ^{3}$ with respect to the basis ${}{\begin{pmatrix}1\\2\\3\end{pmatrix}},\,{\begin{pmatrix}2\\4\\5\end{pmatrix}}$ and ${}{\begin{pmatrix}0\\1\\5\end{pmatrix}}$.

Exercise (2 marks)

Show that the degeneracy space of a symmetric bilinear form ${}\left\langle -,-\right\rangle$ on a ${}K$-vector space ${}V$ equals the kernel of the linear mapping

$V\longrightarrow {V}^{*},v\longmapsto \left\langle v,-\right\rangle .$

Exercise (3 (1+1+1) marks)

We consider the linear form

$L\colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ,(x,y)\longmapsto 4x+7y.$

a) Determine the left gradient of ${}L$ with respect to the determinant.

b) Determine the right gradient of ${}L$ with respect to the determinant.

c) Determine the gradient of ${}L$ with respect to the standard inner product.

Exercise (3 marks)

Let ${}V$ be a ${}K$-vector space. Show that the set of all symmetric bilinear forms on ${}V$ forms a linear subspace of the space of all bilinear forms. What is the dimension of this space, if

${}\dim _{K}{\left(V\right)}=n\,?$

Exercise (1 mark)

Let ${}V$ be a real vector space. Does the set of all inner products on ${}V$ form a linear subspace of the space of all bilinear forms on ${}V$?

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