Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheet 31

Exercises

Exercise

Show that the standard inner product on ${}\mathbb {R} ^{n}$ is indeed an inner product.

Exercise

Let ${}V$ be a ${}{\mathbb {K} }$-vector space, endowed with an inner product ${}\left\langle -,-\right\rangle$, and let ${}U\subseteq V$ denote a linear subspace. Show that the restriction of the inner product to ${}U$ is again an inner product.

Exercise

Let ${}V$ be a complex vector space, endowed with an inner product ${}\left\langle -,-\right\rangle$. Show that the real part of this inner product is an inner product on the underlying real vector space.

Exercise

Let ${}\left\langle -,-\right\rangle$ denote a bilinear form on ${}\mathbb {R} ^{2}$, with the values ${}\left\langle e_{1},e_{1}\right\rangle =5$, ${}\left\langle e_{1},e_{2}\right\rangle =-4$, ${}\left\langle e_{2},e_{1}\right\rangle =7$, and ${}\left\langle e_{2},e_{2}\right\rangle =-3$. Compute ${}\left\langle {\begin{pmatrix}3\\-4\end{pmatrix}},{\begin{pmatrix}-5\\-6\end{pmatrix}}\right\rangle$. Is it an inner product?

Exercise

Let

$f,g\colon [0,1]\longrightarrow \mathbb {R}$

be given by ${}f(x)=-x^{2}+1$, and ${}g(x)=x^{2}+x+3$. Compute ${}\left\langle f,g\right\rangle$ in the sense of Example 31.6 .

Exercise

Let ${}[a,b]$ be a closed real interval with ${}a<b$, and let ${}V={\left\{f:[a,b]\rightarrow \mathbb {R} \mid f{\text{ continuous}}\right\}}$. For ${}n\in \mathbb {N} _{+}$ and ${}f,g\in V$, let

${}\left\langle f,g\right\rangle _{n}:=\sum _{i=0}^{n}f{\left(a+i{\frac {b-a}{n}}\right)}g{\left(a+i{\frac {b-a}{n}}\right)}\,.$

What properties of an inner product

does ${}\left\langle -,-\right\rangle _{n}$ satisfy? What relation exists between ${}\left\langle -,-\right\rangle _{n}$ and the inner product from Example 31.6 ?

Exercise

Let ${}(V_{1},\left\langle -,-\right\rangle _{1})$ and ${}(V_{2},\left\langle -,-\right\rangle _{2})$ be real vector spaces, endowed with inner products. Show that by

${}\left\langle (v_{1},v_{2}),(w_{1},w_{2})\right\rangle :=\left\langle v_{1},w_{1}\right\rangle _{1}+\left\langle v_{2},w_{2}\right\rangle _{2}\,,$

an inner product is defined on the product space ${}V_{1}\times V_{2}$.

Exercise

Let ${}V$ denote a vector space over ${}{\mathbb {K} }$, endowed with an inner product ${}\left\langle -,-\right\rangle$. Let ${}\Vert {-}\Vert$ denote the associated norm. Show that the so-called Parallelogram law holds, that is,

${}\Vert {v+w}\Vert ^{2}+\Vert {v-w}\Vert ^{2}=2\Vert {v}\Vert ^{2}+2\Vert {w}\Vert ^{2}\,.$

Exercise

Let ${}V$ be a real vector space, endowed with an inner product ${}\left\langle -,-\right\rangle$. Show that in the estimate

${}\vert {\left\langle v,w\right\rangle }\vert \leq \Vert {v}\Vert \cdot \Vert {w}\Vert \,$

of Cauchy-Schwarz, equality holds if and only if ${}v$ and ${}w$ are linearly dependent.

Exercise

Let ${}V$ denote a vector space over ${}{\mathbb {K} }$, endowed with an inner product ${}\left\langle -,-\right\rangle$. Let ${}\Vert {-}\Vert$ denote the associated norm.

a) Show that, in the case ${}{\mathbb {K} }=\mathbb {R}$, the relation

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

holds.

b) Show that, in the case ${}{\mathbb {K} }=\mathbb {C}$, the relation

$\left\langle v,w\right\rangle ={\frac {1}{4}}{\left(\Vert {v+w}\Vert ^{2}-\Vert {v-w}\Vert ^{2}+{\mathrm {i} }\Vert {v+{\mathrm {i} }w}\Vert ^{2}-{\mathrm {i} }\Vert {v-{\mathrm {i} }w}\Vert ^{2}\right)}\,$

holds.

Exercise

Let ${}V$ be a complex vector space, endowed with an inner product ${}\left\langle -,-\right\rangle$. Show that the norm of this inner product coincides with the norm arising by considering ${}V$ as a real vector space, endowed with the corresponding real inner product.

Exercise

Show that the maximum norm on ${}{\mathbb {K} }^{n}$ is a norm.

Exercise

Show that the sum norm on ${}{\mathbb {K} }^{n}$ is a norm.

Exercise

Determine, for the vector

${}{\begin{pmatrix}-6\\10\\7\end{pmatrix}}\in \mathbb {R} ^{3}\,,$

the corresponding normed vector with respect to the Euclidean norm, the maximum norm, and the sum norm.

Exercise

Let ${}n\geq 2$. Show that the norm ${}\Vert {x}\Vert :=\operatorname {max} \{\vert {x_{i}}\vert :1\leq i\leq n\}$ on ${}\mathbb {R} ^{n}$ does not come from an inner product ${}\left\langle -,-\right\rangle$.

Exercise

Let ${}V$ be a normed vector space over ${}\mathbb {C}$. Show that ${}V$, considered as a real vector space, is, with this norm, also a normed vector space.

Exercise *

Show that a normed ${}{\mathbb {K} }$-vector space is via

${}d(u,v):=\Vert {u-v}\Vert \,$

also a metric space.

Exercise

We consider the two points ${}P=\left({\frac {3}{4}},\,-1\right)$ and ${}Q=\left(2,\,{\frac {1}{5}}\right)$ in ${}\mathbb {R} ^{2}$. Determine the distance between these points with respect to the

a) the Euclidean metric

b) the sum metric,

c) the maximum metric.

d) Compare these different distances according to their size.

Let ${}(M,d)$ be a metric space, ${}x\in M$, and ${}\epsilon >0$ a positive real number. Then

${}U{\left(x,\epsilon \right)}={\left\{y\in M\mid d(x,y)<\epsilon \right\}}\,$

is called the open ball

about ${}x$ with radius ${}\epsilon$.

Exercise

Let ${}A$ denote a set, ${}n\in \mathbb {N} _{+}$, and ${}M=A^{n}$ the ${}n$-fold product set of ${}A$ with itself.

a) Show that

${}d(x,y)=d((x_{1},\ldots ,x_{n}),(y_{1},\ldots ,y_{n})):={\#\left({\left\{i\mid x_{i}\neq y_{i}\right\}}\right)}\,$

defines a metric on ${}M$.

b) For ${}A=\{a,b,c\}$, and ${}n=4$, determine the distance ${}d((a,a,b,c),(c,a,b,a))$.

c) For ${}A=\{a,b,c\}$, and ${}n=3$, list all elements in the open ball ${}U{\left((a,a,b),2\right)}$.

Exercise

Let ${}M$ be the set of all train stations in Germany. For ${}a,b\in M$, define

$d(a,b)$

to be the shortest (in terms of scheduled time) connection from ${}a$ to ${}b$. Is this a metric?

Exercise

Let ${}A\subseteq \mathbb {R} ^{2}$ be the union of the ${}x$-axis and the ${}y$-axis.

a) Define the distance on ${}A$ given by the shortest connection between the two points ${}P,Q\in A$ by a path on ${}A$.

b) Show that this is a metric.

c) Does there exist a norm on ${}\mathbb {R} ^{2}$ such that the restriction of the corresponding metric coincides with the given connecting metric?

Exercise

Let ${}V$ be a normed ${}{\mathbb {K} }$-vector space, and let ${}E$ denote an affine space over ${}V$. Show that ${}E$ becomes a metric space by setting

${}d(P,Q):=\Vert {\overrightarrow {PQ}}\Vert \,.$

Let ${}T$ be a set, and let

$f\colon T\longrightarrow {\mathbb {K} }$

denote a function. Then

${}\Vert {f}\Vert :=\Vert {f}\Vert _{T}={\operatorname {sup} \,{\left(\vert {f(x)}\vert {|}x\in T\right)}}\,$

is called the supremum (or the supremum norm) of ${}f$. It is a nonnegative real number,

or ${}\infty$.

Exercise

Let ${}T$ be a set, and let

${}M={\left\{f:T\rightarrow \mathbb {C} \mid \Vert {f}\Vert _{T}<\infty \right\}}\,$

denote the set of bounded complex-valued functions on ${}T$. Show that ${}M$ is a complex vector space.

Exercise

Let ${}T$ be a set, and let

${}M={\left\{f:T\rightarrow \mathbb {C} \mid \Vert {f}\Vert _{T}<\infty \right\}}\,$

be the set of all bounded complex-valued functions on ${}T$. Show that the supremum norm on ${}M$ fulfills the following properties.

1. ${}\Vert {f}\Vert \geq 0$ for all ${}f\in M$.
2. ${}\Vert {f}\Vert =0$ if and only if ${}f=0$.
3. For ${}\lambda \in \mathbb {C}$, and ${}f\in M$, we have

   ${}\Vert {\lambda f}\Vert =\vert {\lambda }\vert \cdot \Vert {f}\Vert \,.$

4. For ${}g,f\in M$, we have

   ${}\Vert {g+f}\Vert \leq \Vert {g}\Vert +\Vert {f}\Vert \,.$

Exercise

Let ${}T$ be a set, and let ${}E$ be a Euclidean vector space. Let

${}M={\left\{f:T\rightarrow E\mid \Vert {f}\Vert _{T}<\infty \right\}}\,$

be the set of all bounded mappings from ${}T$ to ${}E$. Show that the supremum norm on ${}M$ is a norm.

Exercise

Let ${}T$ be a set, and let ${}E$ be a Euclidean vector space. Let

${}M={\left\{f:T\rightarrow E\mid \Vert {f}\Vert _{T}<\infty \right\}}\,$

be the set of all bounded mappings from ${}T$ to ${}E$. Show that a sequence ${}{\left(f_{n}\right)}_{n\in \mathbb {N} }$ in ${}M$ converges uniformly to ${}f\in M$ if and only if this sequence converges to ${}f$ in ${}M$ with respect to the supremum norm.

Hand-in-exercises

Exercise (4 marks)

Let

$f,g\colon [0,1]\longrightarrow \mathbb {R}$

with ${}f(x)=2x^{3}-x+3$ and ${}g(x)=-5x^{2}+4x-7$. Compute ${}\left\langle f,g\right\rangle$ in the sense of Example 31.6 .

Exercise (2 marks)

Let ${}V$ be a real vector space, endowed with an inner product ${}\left\langle -,-\right\rangle$. Confirm the identity

${}\Vert {x+y}\Vert ^{2}-\Vert {x-y}\Vert ^{2}=4\left\langle x,y\right\rangle \,.$

Exercise (4 marks)

Let ${}V=\operatorname {Mat} _{n}(\mathbb {R} )$. Show that ${}V$, endowed with the mapping

$\left\langle -,-\right\rangle \colon V\times V\longrightarrow \mathbb {R} ,(A,B)\longmapsto \operatorname {trace} {\left(B^{t}A\right)},$

is a Euclidean vector space.

Exercise (5 marks)

Let ${}n$ points ${}P_{1},P_{2},\ldots ,P_{n}$ in the unit disk ${}B$ be given, that is, ${}B={\left\{P\in \mathbb {R} ^{2}\mid d(P,0)\leq 1\right\}}$. Show that there exists a point ${}Q\in B$ fulfilling the property

${}\sum _{i=1}^{n}d(P_{i},Q)\geq n\,.$

Exercise (3 marks)

Let ${}v,w\in \mathbb {R} ^{n}$ be vectors fulfilling ${}\Vert {v}\Vert \leq 1$ and ${}\Vert {w}\Vert =1$. Show that there exists an ${}a\in \mathbb {R}$ such that ${}\Vert {v+aw}\Vert =1$ holds.

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