Null Spaces and Ranges Exercise 1 Give an example of a linear map \( T \) such that \(\dim \text{null} \, T = 3\) and \(\dim \text{range} \, T = 2\). Exercise 2 Suppose \( V \) is a vector space and \( S, T \in \mathcal{L}(V, V) \) are such that \[\text{range} \, S \subset \text{null} \, T.\] Prove that \((ST)^2 = 0\). Exercise 3 Suppose \( v_1, \ldots, v_m \) is a list of vectors in \( V \). Define \( T \in \mathcal{L}(\mathbb{F}^m, V) \) by ...
Vector Space of Linear Maps Exercise 1 Suppose \( b, c \in \mathbb{R} \). Define \( T: \mathbb{R}^3 \to \mathbb{R}^2 \) by \[T(x, y, z) = (2x - 4y + 3z + b, 6x + cxyz).\] Show that \( T \) is linear if and only if \( b = c = 0 \). Exercise 2 Suppose \( b, c \in \mathbb{R} \). Define \( T: \mathcal{P}(\mathbb{R}) \to \mathbb{R}^2 \) by ...
Dimension Exercise 1 Suppose \( V \) is finite-dimensional and \( U \) is a subspace of \( V \) such that \[\dim U = \dim V \] Prove that \( U = V \) Exercise 2 Show that the subspaces of \( \mathbb{R}^2 \) are precisely \(\{0\}, \mathbb{R}^2\), and all lines in \( \mathbb{R}^2 \) through the origin. Exercise 3 Show that the subspaces of \( \mathbb{R}^3 \) are precisely \(\{0\}, \mathbb{R}^3\), all lines in \( \mathbb{R}^3 \) through the origin, and all planes in \( \mathbb{R}^3 \) through the origin. ...
Bases Exercise 1 Find all vector spaces that have exactly one basis. Exercise 2 Verify all the assertions: The list \((1,0,\ldots,0), (0,1,0,\ldots,0), \ldots, (0,\ldots,0,1)\) is a basis of \(\mathbb{F}^n\), called the standard basis of \(\mathbb{F}^n\). The list \((1,2), (3,5)\) is a basis of \(\mathbb{F}^2\). The list \((1,2,-4), (7,-5,6)\) is linearly independent in \(\mathbb{F}^3\) but is not a basis of \(\mathbb{F}^3\) because it does not span \(\mathbb{F}^3\). The list \((1,2), (3,5), (4,13)\) spans \(\mathbb{F}^2\) but is not a basis of \(\mathbb{F}^2\) because it is not linearly independent. ...
Subspaces Exercise 1 For each of the following subsets of $\mathbb{F}^3$, determine whether it is a subspace of $\mathbb{F}^3$: ${(x_1, x_2, x_3) \in \mathbb{F}^3 : x_1 + 2x_2 + 3x_3 = 0}$ ${(x_1, x_2, x_3) \in \mathbb{F}^3 : x_1 + 2x_2 + 3x_3 = 4}$ ${(x_1, x_2, x_3) \in \mathbb{F}^3 : x_1 x_2 x_3 = 0}$ ${(x_1, x_2, x_3) \in \mathbb{F}^3 : x_1 = 5x_3}$ Exercise 2 Show that the set of differentiable real-valued functions $f$ on the interval $(-4, 4)$ such that $f’( -1) = 3f(2)$ is a subspace of $\mathbb{R}^{(-4, 4)}$. ...
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 $\mathbb{R}^2$. ...