# Category Archives: Chapter 1

## Chapter 1 Exercise C

1. Solution: (a) $\{(x_1,x_2,x_3)\in\mathbb F^3:x_1+2x_2+3x_3=0\}$ is a subspace of $\mathbb F^3$. By 1.34, to show a subset is a subspace, we just need to check Additive identity, Closed under addition and Closed under scalar multiplication. Additive identity: it is clear … Continue reading

## Chapter 1 Exercise B

1. Prove that $-(-v)=v$ for every $v\in V$. Solution: By definition, we have\[(-v)+(-(-v))=0\quad\text{and}\quad v+(-v)=0.\]This implies both $v$ and $-(-v)$ are additive inverses of $-v$, by the uniqueness of additive inverse, it follows that $-(-v)=v$. 2. Suppose $a\in\mathbb F$, $v\in V$, … Continue reading

## Chapter 1 Exercise A

1. Suppose $a$ and $b$ are real numbers, not both 0. Find real numbers $c$ and $d$ such that\[\frac{1}{a+bi}=c+di.\]Solution: Because $(a+bi)(a-bi)=a^2+b^2$, one has\[\frac{1}{a+bi}=\frac{a-bi}{a^2+b^2}.\]Hence\[c=\frac{a}{a^2+b^2},d=-\frac{b}{a^2+b^2}.\] 2. Show that \[\frac{-1+\sqrt{3}i}{2}\] is a cube root of 1 (meaning that its cube equals 1). Soltion1:From … Continue reading