Problem Let $\infty$ and $-\infty$ denote two distinct objects, neither of which is in $\mathbb{R}$. Define an addition and scalar multiplication on $R \cup {\infty} \cup {-\infty}$ as you could guess from the notation. Specifically, the sum and product of two real numbers is as usual, and for $t \in \mathbb{R}$ define Is $\mathbb{R} \cup {\infty} \cup {-\infty}$ a vector space over $\mathbb{R}$? Explain. Solution No, because: $$1 = 1 + 0 = 1 + (\infty - \infty) \stackrel{D1....

Problem Show that in the definition of a vector space (1.19), the additive inverse condition can be replaced with the condition that $$0v = 0 \textrm{ for all } v \in V$$ Here the $0$ on the left side is the number $0$, and the $0$ on the right side is the additive identity of $V$ Solution Let $v,w \in V$, such then: $$0 = 0v = (1 + (-1))v \stackrel{D1....

Problem The empty set is not a vector space. The empty set fails to satisfy only one of the requirements listed in 1.19. Which one? Solution Definition 1.19 requires that there exists an additive identity $0 \in V$, but as there are no elements in the empty set, this requirement is not satisfied by the empty set.

Problem Suppose $v, w \in V$. Explain why there exists a unique $x \in V$ such that $v + 3x = w$ Solution Suppose that $x,x^\prime \in V$ satisfies $v + 3x = w = v + 3x^\prime$, then adding the additive inverse of $v$ to both sides will reduce the equality to $3x = 3x^\prime$, dividing by $3$ on both sides results in $x = x^\prime$, thus $x$ is unique....

Problem Suppose $a \in \mathbb{F}, v \in V$, and $av = 0$. Prove $a = 0$ or $v = 0$ Solution First suppose $a \neq 0$, then: $$0 \stackrel{T1.30}{=} \frac{1}{a}0 = \frac{1}{a}(av) \stackrel{P1A13}{=} \left(\frac{1}{a} \cdot a\right)v = 1v \stackrel{D1.19}{=} v$$ Now suppose $v \neq 0$, then any vector $u \in V$ can be written as $w + v$ for $w \in V$, since $w = u - v \to u = w + v$, but then: $$au = a(w + v) \stackrel{D1....