Axle - 4th : Chapter 1

Disclaimer: Proofs are not of their most formal forms. The notation $0$ could refer to either a scalar or the zero vector.

Exercise 1A

Problems 1 - 5

Simple derivations.

Problem 6

Show that for every $\alpha \in \mathbb{C}$, there exists a unique $\beta \in \mathbb{C}$ such that $\alpha \beta = 1$.

Proof. W.l.o.g., assume that $\alpha = a + bi \neq 0$, as $0$ has no multiplicative inverse. We first show the uniqueness of an inverse, assuming its existence. Suppose

It follows that

To prove the existence of an inverse, one can verify that, with

we have $(a + bi) (c + di) = 1$. Since $a + bi \neq 0 \Rightarrow a^2 + b^2 \neq 0$, thus $\beta = c + di$ exists. This concludes the proof.

Problem 7

Show that $(-1 + \sqrt{3}i)/2$ is a cube root of $1$.

Ans. Either via a direct calculation, or the fact that $1$ has three cube roots: $e^{(\pi/3) \cdot (2k+1)}$, $k = 0, 1, 2$.

Problem 8

Find two distinct square roots of $i$.

Ans. $e^{\pi/4 \cdot i}$ and $e^{{5\pi/4 \cdot i}}$.

Remaining Problems

Calculations and derivations.

Exercise 1B

Problem 1

Prove that $-(-v) = v$ for every $v \in V$.

Ans. The additive inverse is unique.

Problem 2

Suppose $a \in \mathbb{F}$, $v \in V$, and $av = 0$. Prove $a = 0$ or $v = 0$.

Proof. If $a = 0$, then $av = 0$. Suppose $a \neq 0$, then there exists $a^{-1} \in \mathbb{F}$ s.t. $a^{-1} a= 1$. Then

This concludes the proof.

Problem 3

Derivations.

Problem 4

Why the empty set is not a vector space.

Ans. An additive identity must exist in a vector space.