Span and Linear Independence Exercise 1 Suppose $v_1, v_2, v_3, v_4$ spans $V$. Prove that the list $$v_1 - v_2, v_2 - v_3, v_3 - v_4, v_4$$ also spans $V$. Exercise 2 Verify the assertions: A list $v$ of one vector $v \in V$ is linearly independent if and only if $v \neq 0$. A list of two vectors in $V$ is linearly independent if and only if neither vector is a scalar multiple of the other. ...
Preliminary Analysis Exercise 1 Prove that $\sqrt{3}$ is irrational. Does a similar argument work to show $\sqrt{6}$ is irrational? Exercise 2 Decide which of the following represent true statements about the nature of sets. For any that are false, provide a specific example where the statement in question does not hold. If $ A_1 \supseteq A_2 \supseteq A_3 \supseteq A_4 \supseteq \cdots $ are all sets containing an infinite number of elements, then the intersection $ \bigcap_{n=1}^{\infty} A_n $ is infinite as well. If $ A_1 \supseteq A_2 \supseteq A_3 \supseteq A_4 \supseteq \cdots $ are all finite, nonempty sets of real numbers, then the intersection $ \bigcap_{n=1}^{\infty} A_n $ is finite and nonempty. $A \cap (B\cup C) = (A \cap B )\cup C$ $A \cap (B\cap C) = (A \cap B )\cap C$ $A \cap (B\cup C) = (A \cap B ) \cup (A \cap C)$ Exercise 3: De Morgan’s Law Let $ A $ and $ B $ be subsets of $ \mathbb{R} $. ...
Axiom Of Completeness Exercise 1 Let $ \mathbb{Z}_5 = {0, 1, 2, 3, 4} $ and define addition and multiplication modulo 5. In other words, compute the integer remainder when $ a + b $ and $ ab $ are divided by 5, and use this as the value for the sum and product, respectively. Show that, given any element $ z \in \mathbb{Z}_5 $, there exists an element $ y $ such that $ z + y = 0 $. The element $ y $ is called the additive inverse of $ z $. Show that, given any $ z \neq 0 $ in $ \mathbb{Z}_5 $, there exists an element $ x $ such that $ zx = 1 $. The element $ x $ is called the multiplicative inverse of $ z $. The existence of additive and multiplicative inverses is part of the definition of a field. Investigate the set $ \mathbb{Z}_4 = {0, 1, 2, 3} $ (where addition and multiplication are defined modulo 4) for the existence of additive and multiplicative inverses. Make a conjecture about the values of $ n $ for which additive inverses exist in $ \mathbb{Z}_n $, and then form another conjecture about the existence of multiplicative inverses. Exercise 2 Write a formal definition in the style of Definition 1.3.2 for the infimum or greatest lower bound of a set. Now, state and prove a version of Lemma 1.3.7 for greatest lower bounds. Exercise 3 Let $ A $ be bounded below, and define $$ B = \{ b \in \mathbb{R} : b \text{ is a lower bound for } A \}. $$ Show that $ \sup B = \inf A $. Use (1) to explain why there is no need to assert that greatest lower bounds exist as part of the Axiom of Completeness. Propose another way to use the Axiom of Completeness to prove that sets bounded below have greatest lower bounds. Exercise 4 Assume that $A$ and $B$ are nonempty, bounded above, and satisfy $B \subseteq A$. Show that $ \sup(B) \leq \sup(A) $. ...
Eigenvalues and Eigenvectors Exercise 1 Confirm by multiplication that x is an eigenvector of A, and find the corresponding eigenvalue. \( A = \begin{bmatrix} 1 & 2 \\ 3 & 2 \end{bmatrix}, \quad x = \begin{bmatrix} 1 \\ -1 \end{bmatrix} \) \( A = \begin{bmatrix} 5 & -1 \\ 3 & 2 \end{bmatrix}, \quad x = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \) \( A = \begin{bmatrix} 4 & 0 & 0 \\ 2 & 3 & 2 \\ 1 & 0 & 4 \end{bmatrix}, \quad x = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \) \( A = \begin{bmatrix} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{bmatrix}, \quad x = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \) Exercise 2 Find the characteristic equation, the eigenvalues, and bases for the eigenspaces of the matrix. ...
Inner Product Spaces Exercise 1 In each part, determine whether the set of vectors is orthogonal and whether it is orthonormal with respect to the Euclidean inner product on ℝ². $ (0, 1), (2, 0) $ $(- \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}), (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$ $(- \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}), (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$ $ (0, 0), (0, 1) $ Exercise 2 In each part, determine whether the set of vectors is orthogonal and whether it is orthonormal with respect to the Euclidean inner product on $\mathbb{R}^2$. ...
Inner Products Exercise 1 Let $\mathbb{R}^2$ have the weighted Euclidean inner product: $$ \langle u, v \rangle = 2u_1v_1 + 3u_2v_2 $$and let $u = (1, 1)$, $v = (3, 2)$, $w = (0, -1)$, and $k = 3$. Compute the stated quantities. $ \langle u,v \rangle $ $ \langle ku,w \rangle $ $ \langle u+v , w \rangle $ $ | v | $ $ d\langle u,v \rangle $ $ | u-kv | $ Exercise 2 Let $\mathbb{R}^2$ have the weighted Euclidean inner product: ...
Fonctions de plusieurs variables Exercice 1 : Continuité de fonctions de deux variables Étudier la continuité en $(0,0)$ des fonctions suivantes : $$f(x, y) = \frac{xy(x^2 - y^2)}{x^2 + y^2}, \quad f(0,0) = 0.$$ $$f(x, y) = \frac{x^4 + y^4}{x^2 + y^2}, \quad f(0,0) = 0.$$ $$f(x, y) = \frac{x^2 y^2}{x^2 + y^2}, \quad f(0,0) = 0.$$ $$f(x, y) = \frac{xy^2}{x^2 + y^4}, \quad f(0,0) = 0.$$ $$f(x, y) = \frac{\sin{|xy|}}{x^2 + y^4}, \quad f(0,0) = 0.$$ $$f(x, y) = \frac{x^4y^4}{x^2 + 2xy + y^2}, \quad f(0,0) = 0.$$ $$f(x, y) = \frac{x^2 + 2xy + y^2}{x^2 + y^2}, \quad f(0,0) = 0.$$ Exercice 2 : Différentiabilité de fonctions de deux variables Étudier la continuité et la différentiabilité des fonctions suivantes, puis lorsqu’elles sont différentiables, donner l’équation de leur plan tangent à la surface d’équation $z = f(x, y)$ en $(1,0,f(1,0))$ : ...
Séries numériques Exercice 1 : Calculs de séries Calculer les sommes des séries suivantes : $$ \sum_{k=0}^{+\infty} \left(\frac{2}{3}\right)^k $$ $$ \sum_{k=0}^{+\infty} 3^{-2k+1} $$ $$ \sum_{k=0}^{+\infty} \sin(2ka)e^{-ka} $$ $$ \sum_{k=0}^{+\infty} \frac{k}{2^k} \quad \text{(plus difficile).} $$ Exercice 2 : Convergence de séries Étudier la convergence de la série $\sum_{n=1}^{+\infty} u_n$, pour : $$ u_n = \frac{n}{n^2 + 1} $$ $$ u_n = \frac{\cosh n}{\cosh(2n)} $$ $$ u_1 = 0 \quad \text{et} \quad u_n = \frac{1}{\sqrt{n^2-1}} - \frac{1}{\sqrt{n^2+1}} \quad \text{pour } n > 1 $$ $$ u_n = \frac{n!}{n^n} $$ $$ u_n = e - \left(1 + \frac{1}{n}\right)^n $$ $$ u_n = \left( \frac{n}{n+1} \right)^{n^2} $$ $$ u_n = \int_0^1 \sin \left( \frac{t}{n} \right) dt $$ $$ u_n = \frac{1}{\cos^2 n} $$ Exercice 3 : Convergence absolue Pour chacune des séries suivantes, dites si elle converge absolument et si les sommes partielles convergent. ...
Convergence et calcul d’intégrales impropres Exercice 1 : Convergence et calcul d’intégrales impropres Étudier la convergence des intégrales impropres suivantes et, lorsqu’elles sont convergentes, calculer leurs valeurs. $$ \int_1^3 \frac{dt}{\sqrt{(t-1)(3-t)}} $$ Pour $ b > 0 $. $$ \int_0^b \ln(t) dt $$ $$ \int_1^{+\infty} \frac{e^{1/t}}{t^2} dt $$ $$ \int_0^x \frac{t \ln(t)}{(t^2+1)^2} dt $$ Exercice 2 : Convergences sans calcul Étudier la convergence des intégrales impropres suivantes : $$ \int_1^{+\infty} e^{-t^2} dt $$ $$ \int_0^\infty \frac{\sin t}{1+t^2} dt $$ $$ \int_0^1 \frac{dt}{\ln t} $$ En fonction de $ \alpha \in \mathbb{R} $ ...
Équations Différentielles Exercice 1 : Applications Directes Résoudre les équations différentielles suivantes : $ xy’ + 2y = 4x^2 $, avec la condition initiale $ y(1) = 2 $. $ y’ \sin(x) - y = 1 - \cos x $. $ (1 - x^2)y’ + xy = 1 $. $ xy’ + y = y^2 \log(x) $. $ y’ + 2xy + xy^4 = 0 $. $ y - xy’ + \log x = 0 $. $ x^3y’ + (2 - 3x^2)y = x^3 $. $ y’ + y = y^2 (\cos x - \sin x) $. Exercice 2 : Équations du premier ordre à variable séparée On considère les équations de la forme : ...