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 $\mathbb{R}^{(-4, 4)}$. ...
Definition of Vector Space Exercise 1: Double Additive Inverse Prove that $ -(-v) = v $ for every $ v \in V $, where $ V $ is a vector space. Exercise 2: Zero Product Property Suppose $ a \in \mathbb{F} $ (a field), $ v \in V $, and $ av = 0 $. Prove that $ a = 0 $ or $ v = 0 $. Exercise 3: Unique Solution to Vector Equation Suppose $ v, w \in V $. Explain why there exists a unique $ x \in V $ such that $ v + 3x = w $. ...
Léisung zu dësen Exercicer sinn am Archiv op luxformel ze fannen. Complex Numbers and Vector Space Exercise 1 : Complex Number Inverse Suppose $a $ and $b $ are real numbers, not both 0. Find real numbers $c $ and $d $ such that $$ \frac{1}{a + bi} = c + di. $$Exercise 2 : Cube Root of Unity Show that $$ \frac{-1 + \sqrt{3}i}{2} $$ is a cube root of 1 (meaning that its cube equals 1). ...