Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheet 32

Exercises

The following three exercises show that the concept of orthogonality defined via an inner product coincides with the elementary-geometric concept of orthogonality.

Exercise

Let $V$ be a $𝕂$-vector space, endowed with an inner product $⟨-,-⟩$. Show that, for $u,v\in V$, the equality $\Vert v+u\Vert =\Vert v-u\Vert$ holds if and only if $⟨v,u⟩=0$ holds.

Exercise

Show in an elementary-geometric way that the vectors $\left(\begin{array}{c}a\\ b\end{array}\right)$ and $\left(\begin{array}{c}-b\\ a\end{array}\right)$ in ${ℝ}^2$ are perpendicular to each other.

Exercise

Let

$v=\left(\begin{array}{c}{a}_1\\ a_2\\ ⋮\\ a_n\end{array}\right)\in {ℝ}^n$

be a vector.

a) Show in an elementary-geometric way that every vector of the form

$v_{ij}=-a_j{e}_i+a_i{e}_j$

for $i\ne j$ is perpendicular to $v$.

b) Show that every vector

$w=\left(\begin{array}{c}{b}_1\\ b_2\\ ⋮\\ b_n\end{array}\right)\in {ℝ}^n$

satisying

$\sum _{i=1}^n{a}_i{b}_i=0$

is a linear combination of the $v_{ij}$ (and is, therefore, perpendicular to $v$).

c) Show that a vector $w$ that is perpendicular to $v$ in the elementary-geometric sense fulfills the condition $\sum _{i=1}^n{a}_i{b}_i=0$.

Exercise

Determine which of the following vectors in ${ℝ}^3$ are orthogonal to each other with respect to the standard inner product.

$\left(\begin{array}{c}6\\ 1\\ 5\end{array}\right),\left(\begin{array}{c}3\\ -8\\ -2\end{array}\right),\left(\begin{array}{c}0\\ -1\\ 4\end{array}\right),\left(\begin{array}{c}-5\\ 4\\ -1\end{array}\right).$

Exercise

Determine which of the following vectors in ${ℂ}^2$ are orthogonal to each other with respect to the standard inner product.

$\left(\begin{array}{c}6-3\mathrm{i}\\ -\mathrm{i}\end{array}\right),\left(\begin{array}{c}4-7\mathrm{i}\\ -9-5\mathrm{i}\end{array}\right),\left(\begin{array}{c}\frac{1}{3}\mathrm{i}\\ 2+\mathrm{i}\end{array}\right),\left(\begin{array}{c}\frac{1}{3}\\ -1-2\mathrm{i}\end{array}\right),\left(\begin{array}{c}\frac{1}{3}\\ -1+2\mathrm{i}\end{array}\right).$

Exercise

Let $V$ be a $𝕂$-vector space, endowed with an inner product $⟨-,-⟩$. Let $v\in V$ be a fixed vector, and $a\in 𝕂$. Show that

$\{u\in V\mid ⟨u,v⟩=a\}$

is an affine subspace of $V$.

Exercise

Discuss the relationship between the Pythagorean theorem in the sense of Theorem 32.3 and its elementary-geometric version.

Exercise

Let $V$ be a vector space over $𝕂$, endowed with an inner product $⟨-,-⟩$, and let $U\subseteq V$ be a linear subspace. Show that the orthogonal complement is also a linear subspace of $V$.

Exercise

Let $V$ be a vector space over $𝕂$, endowed with an inner product $⟨-,-⟩$, and let $U\subseteq V$ be a linear subspace. Let $u_i$, $i\in I$, be a generating system of $U$. Show that a vector $v\in V$ belongs to the orthogonal complement $U^{\perp }$ if and only if

$⟨v,u_i⟩=0$

holds for all $i\in I$.

Exercise

Determine the orthogonal complement of the linear subspace in ${ℝ}^3$ generated by $\left(\begin{array}{c}-2\\ 8\\ 9\end{array}\right)$.

Exercise

Determine the orthogonal complement of the linear subspace in ${ℝ}^4$ generated by $\left(\begin{array}{c}5\\ 8\\ -3\\ 9\end{array}\right)$ and $\left(\begin{array}{c}6\\ 2\\ 0\\ 3\end{array}\right)$.

Exercise

Let $V$ be a vector space over $𝕂$, endowed with an inner product $⟨-,-⟩$. Let $v_i$, $i\in I$, denote an orthogonal basis of $V$. For every subset $J\subseteq I$, denote the linear subspace generated by $v_i$, $i\in J$, by $U_J$. Show that the orthogonal complement of $U_J$ equals $U_{I\setminus J}$.

Exercise

We consider a corner in a (rectangular) room. Do the three diagonal vectors in the two adjacent walls and on the ground of length $1$ form an orthonormal basis?

Exercise

Let $V$ be a finite-dimensional complex vector space, endowed with an inner product $⟨-,-⟩$. Let $u_1,\dots ,u_n$ be an orthonormal basis of $V$. Show that

$u_1,\mathrm{i}{u}_1,u_2,\mathrm{i}{u}_2,\dots ,u_n,\mathrm{i}{u}_n$

is an orthonormal basis of the real vector space $V$ with respect to the corresponding real inner product.

Exercise

Let $V$ be a real vector space, endowed with an inner product $⟨-,-⟩$. Let $u_i$, $i\in I$, denote an orthonormal basis of $V$. Show that for vektors $v=\sum _{i\in I}{a}_i{u}_i$ and $w=\sum _{i\in I}{b}_i{u}_i$, the equality

$⟨v,w⟩=\sum _{i\in I}{a}_i{b}_i$

holds.

The following exercise is the foundation of the so-called Fourier-analysis; its purpose is to represent oscillations as an infinite sum of trigonometric (harmonic) oscillations.

Exercise

Show that the functions

$f_m:[0,1]⟶ℂ,$

defined by

$f_m(x):=e^{2\pi \mathrm{i}mx}$

for $m\in \mathbb{Z}$, form, in the space of continuous functions from $[0,1]$ to $ℂ$, an orthonormal system with respect to the inner product defined by

$⟨f,g⟩:={\int }_0^{1}f\bar{g}dx.$

Use basic facts about the complex exponential function.

Exercise

Apply Gram-Schmidt orthonormalization to the basis

$\left(\begin{array}{c}0\\ 1\\ 1\end{array}\right),\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right),\left(\begin{array}{c}1\\ 1\\ 0\end{array}\right)$

of ${ℝ}^3$.

Exercise

Apply Gram-Schmidt orthonormalization to the basis

$\left(\begin{array}{c}2\\ 1\\ 0\end{array}\right),\left(\begin{array}{c}1\\ 0\\ -1\end{array}\right),\left(\begin{array}{c}0\\ 0\\ 3\end{array}\right)$

of ${ℝ}^3$.

Exercise

Apply Gram-Schmidt orthonormalization to the basis

$\left(\begin{array}{c}-1\\ 2\\ 3\end{array}\right),\left(\begin{array}{c}2\\ -4\\ 5\end{array}\right),\left(\begin{array}{c}7\\ 3\\ 1\end{array}\right)$

of ${ℝ}^3$, endowed with the standard inner product.

Exercise

Apply Gram-Schmidt orthonormalization to the basis

$\left(\begin{array}{c}4-\mathrm{i}\\ 3\end{array}\right),\left(\begin{array}{c}\mathrm{i}\\ 2-\mathrm{i}\end{array}\right)$

of ${ℂ}^2$.

Exercise

Establish an orthonormal basis of ${ℝ}^3$ that contains a multiple of $\left(\begin{array}{c}1\\ 1\\ 1\end{array}\right)$.

Exercise

We endow ${ℝ}^3$ with the standard inner product. Let $U\subseteq {ℝ}^3$ be the kernel of the linear mapping

${ℝ}^3⟶ℝ,(x,y,z)⟼3x+y+7z,$

endowed with the restricted inner product. Determine an orthonormal basis for $U$.

Exercise

We endow ${ℝ}^4$ with the standard inner product. Let $U\subseteq {ℝ}^4$ be the kernel of the linear mapping

${ℝ}^4⟶ℝ,(x,y,z,w)⟼4x-3y+2z-5w.$

Determine an orthonormal basis for $U$.

Exercise

Formulate and prove the "orthonormal basis completion theorem“.

Exercise

Let $V$ be a $𝕂$-vector space, endowed with an inner product, and let $U_1,U_2\subseteq V$ denote linear subspaces. Show that for the orthogonal complements, the equality

${(U_1\cap U_2)}^{\perp }=U_1^{\perp }+U_2^{\perp }$

holds.

Exercise

Let $V$ be a $𝕂$-vector space, endowed with an inner product $⟨-,-⟩$. Show the following statements.

1. For a linear subspace $U\subseteq U^{\prime }\subseteq V$, we have

   $U^{\perp }\supseteq U{\text{'}}^{\perp }.$

2. We have $0^{\perp }=V$ and $V^{\perp }=0$.
3. Let $V$ be finite-dimensional. Then we have

   ${\left(U^{\perp }\right)}^{\perp }=U.$

4. Let $V$ be finite-dimensional. Then we have

   ${\dim }_K\left(U^{\perp }\right)={\dim }_K\left(V\right)-{\dim }_K\left(U\right).$

Exercise

Let $V$ be a Euclidean vector space. Show that

$V⟶V^{\ast },v⟼(w\mapsto ⟨v,w⟩),$

gives an isomorphy between $V$ and its dual space $V^{\ast }$.

Exercise

Prove Corollary 32.13 , using Exercise 32.27 and Lemma 15.6 .

Exercise

Determine the value of the vector $\left(\begin{array}{c}3\\ -5\\ -3\end{array}\right)\in {ℝ}^3$ under the orthogonal projection to the line generated by $\left(\begin{array}{c}2\\ 4\\ 9\end{array}\right)$.

Exercise

Determine the value of the vector $\left(\begin{array}{c}3\\ 4\\ -2\end{array}\right)\in {ℝ}^3$ under the orthogonal projection to the linear subspace generated by $\left(\begin{array}{c}3\\ 0\\ 6\end{array}\right)$ and $\left(\begin{array}{c}1\\ 7\\ 1\end{array}\right)$.

Exercise

Let $V$ be a finite-dimensional $𝕂$-vector space, endowed with an inner product. Let

$U\subseteq W\subseteq V$

denote linear subspaces. Let $p_U^V$ denote the orthogonal projection from $V$ to $U$. Show

$p_U^V=p_U^W\circ p_W^V.$

Hand-in-exercises

Exercise (3 marks)

Determine the orthogonal complement of the linear subspace in ${ℝ}^4$ generated by $\left(\begin{array}{c}8\\ 3\\ -6\\ -4\end{array}\right)$ and $\left(\begin{array}{c}4\\ -2\\ 7\\ 5\end{array}\right)$.

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

We consider the complex numbers $ℂ$ with the standard inner product $⟨-,-⟩$.

a) Determine for the complex linear subspace of $ℂ$ generated by $4+7\mathrm{i}$ the orthogonal complement with respect to $⟨-,-⟩$.

b) Determine for the complex linear subspace of $ℂ$ generated by $4+7\mathrm{i}$ the orthogonal complement with respect to the real part of $⟨-,-⟩$ (that is, the corresponding real inner product).

c) Determine for the real linear subspace of $ℂ$ generated by $4+7\mathrm{i}$ the orthogonal complement with respect to the real part of $⟨-,-⟩$.

Exercise (5 marks)

Apply Gram-Schmidt orthonormalization to the linearly independent polynomial functions

$1,x,x^2,x^3\in V=\{f:[0,1]\to ℝ\mid f\text{ continuous}\}$

with respect to the inner product described in Example 31.6 .

Exercise (2 marks)

Apply Gram-Schmidt orthonormalization to the basis

$\left(\begin{array}{c}5-2\mathrm{i}\\ 3-3\mathrm{i}\end{array}\right),\left(\begin{array}{c}6-\mathrm{i}\\ -2+4\mathrm{i}\end{array}\right)$

of ${ℂ}^2$.

Exercise (4 marks)

Determine the value of the vector $\left(\begin{array}{c}7\\ 6\\ 8\end{array}\right)\in {ℝ}^3$ under the orthogonal projection to the linear subspace generated by $\left(\begin{array}{c}2\\ 2\\ 5\end{array}\right)$ and $\left(\begin{array}{c}4\\ -1\\ 3\end{array}\right)$.

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