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 direct computation, we have$\left(\frac{-1+\sqrt{3}i}{2}\right)^2=\frac{-1-\sqrt{3}i}{2},$hence $\left(\frac{-1+\sqrt{3}i}{2}\right)^3=\frac{-1-\sqrt{3}i}{2}\cdot\frac{-1+\sqrt{3}i}{2}=1.$This means $\dfrac{-1+\sqrt{3}i}{2}$ is a cube root of 1.

Solution2: Note that $(a+bi)+(a-bi)=2a$ and $(a+bi)(a-bi)=a^2+b^2,$ it follows that $\dfrac{-1+\sqrt{3}i}{2}$ is a root of the quadratic equation $x^2+x+1=0$.
For $\frac{-1+\sqrt{3}i}{2}+\frac{-1-\sqrt{3}i}{2}=-1$ and $\frac{-1+\sqrt{3}i}{2}\frac{-1-\sqrt{3}i}{2}=1.$ Because $x^3-1=(x-1)(x^2+x+1)$, we obtain the conclusion.

3. Find two distinct square roots of $i$ .

Solution: If we know that $i=e^{\pi i/2}$, then the square roots are $e^{\pi i/4}\quad\text{and}\quad e^{(\pi i/2+2\pi i)/2}=e^{5\pi i/4}.$ Note that for any $x\in\mathbb R$, one has $e^{xi}=\cos x+i\sin x$. Then $e^{\pi i/4}=\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}=\frac{\sqrt{2}(1+i)}{2}$ and $e^{5\pi i/4}=\cos\frac{5\pi}{4}+i\sin\frac{5\pi}{4}=\frac{-\sqrt{2}(1+i)}{2}.$ Hence the roots are $\dfrac{\sqrt{2}(1+i)}{2}$ and $-\dfrac{\sqrt{2}(1+i)}{2}$.

Remark: If we don’t know this fact, then we should recall how to solve the roots of $x^8-1=0$ or $x^4+1=0$ since $x^2+i=0$ means $x^4+1=0$.

4. Show that $\alpha+\beta=\beta+\alpha$ for all $\alpha,\beta\in\mathbb C$.

Solution: Let $\alpha=x+yi$ and $\beta=z+wi$, where $x,y,z,w\in\mathbb R$, then $\alpha+\beta=(x+yi)+(z+wi)=(x+z)+(y+w)i.$ Similarly, $\beta+\alpha=(z+wi)+(x+yi)=(z+x)+(w+y)i.$ Because $x+z=z+x$ and $y+w=w+y$, we obtain that $\alpha+\beta=\beta+\alpha$.

5. Show that $(\alpha+\beta)+\lambda=\alpha+(\beta+\lambda)$ for all $\alpha,\beta,\lambda\in\mathbb{C}$.

Solution: Let $\alpha=x_1+y_1i$, $\beta=x_2+y_2i$, $\lambda=x_3+y_3i$, where $x_1,x_2,x_3$ and $y_1,y_2,y_3$ are real numbers. Then \begin{aligned} (\alpha+\beta)+\lambda=&((x_1+x_2)+(y_1+y_2)i)+(x_3+y_3i)\\ =&((x_1+x_2)+x_3)+((y_1+y_2)+y_3)i. \end{aligned} Similarly, $\alpha+(\beta+\lambda)=(x_1+(x_2+x_3))+(y_1+(y_2+y_3))i$. Note that$(x_1+x_2)+x_3=x_1+(x_2+x_3)\quad\text{and}\quad (y_1+y_2)+y_3=y_1+(y_2+y_3),$it follows that $(\alpha+\beta)+\lambda=\alpha+(\beta+\lambda)$.

6. Show that $(\alpha\beta)\lambda=\alpha(\beta\lambda)$ for all $\alpha,\beta,\lambda\in\mathbb{C}$.

Solution: Let $\alpha=x_1+y_1i$, $\beta=x_2+y_2i$, $\lambda=x_3+y_3i$, where $x_1,x_2,x_3$ and $y_1,y_2,y_3$ are real numbers. Then \begin{align*} (\alpha\beta)\lambda=&((x_1x_2-y_1y_2)+(x_1y_2+y_1x_2)i)(x_3+y_3i)\\ =&((x_1x_2-y_1y_2)x_3-(x_1y_2+y_1x_2)y_3)\\&+((x_1x_2-y_1y_2)x_3+(x_1y_2+y_1x_2)y_3)i. \end{align*} Similarly, one has \begin{align*} \alpha(\beta\lambda)=&(x_1+y_1i)((x_2x_3-y_2y_3)+(x_2y_3+y_2x_3)i)\\ =&(x_1(x_2x_3-y_2y_3)-y_1(x_2y_3+y_2x_3))\\&+(x_1(x_2y_3-y_2x_3)+y_1(x_2x_3+y_2y_3))i. \end{align*} It is easy to see$(x_1x_2-y_1y_2)x_3-(x_1y_2+y_1x_2)y_3=x_1(x_2x_3-y_2y_3)-y_1(x_2y_3+y_2x_3)$and$(x_1x_2-y_1y_2)x_3+(x_1y_2+y_1x_2)y_3=x_1(x_2x_3-y_2y_3)-y_1(x_2y_3+y_2x_3),$ hence we deduce that $(\alpha\beta)\lambda=\alpha(\beta\lambda)$.

7. Show that for every $\alpha\in\mathbb C$, there exists a unique $\beta\in\mathbb C$ such that $\alpha+\beta=0$.

Solution: Let $\alpha=x_1+y_1i$, $\beta=x_2+y_2i$, where $x_1,x_2$ and $y_1,y_2$ are real numbers. If $\alpha+\beta=0$, then $0=\alpha+\beta=(x_1+x_2)+(y_1+y_2)i.$This means $x_2=-x_1$ and $y_2=-y_1$, which implied uniqueness. If $\beta=-x_1-y_1i$, we also have $\alpha+\beta=0$, which implies existence.

8. Show that for every $\alpha\in\mathbb C$ with $\alpha\ne0$, there exists a unique $\beta\in\mathbb C$ such that $\alpha\beta=1$.

Solution: We already know the existence in Problem 1. Now let us show the uniqueness, if $\alpha\beta=1$, then$\beta=1\cdot\beta=\left(\frac{1}{\alpha}\cdot\alpha\right)\cdot\beta=\frac{1}{\alpha}\cdot\left(\alpha\cdot\beta\right)=\frac{1}{\alpha}\cdot1=\frac{1}{\alpha}.$Here the third equality follows from Problem 6.

9. Show that $\lambda(\alpha+\beta)=\lambda\alpha+\lambda\beta$ for all $\lambda,\alpha,\beta\in\mathbb C$.

Solution: Suppose $\alpha=x_1+y_1i$, $\beta=x_2+y_2i$, $\lambda=a+bi$, where $x_1,x_2,a$ and $y_1,y_2,b$ are real numbers. Then \begin{align*} \lambda(\alpha+\beta)=&(a+bi)((x_1+x_2)+(y_1+y_2)i)\\ =&(a(x_1+x_2)-b(y_1+y_2))+(a(y_1+y_2)+b(x_1+x_2))i\\ =&[(ax_1-by_1)+(ay_1+bx_1)i]+[(ax_2-by_2)+(ay_2+bx_2)i]\\ =&\lambda\alpha+\lambda\beta. \end{align*}

10. Find $x\in\mathbb{R}^4$ such that$(4,-3,1,7)+2x=(5,9,-6,8).$

Solution: Because $(4,-3,1,7)+2x=(5,9,-6,8)$, one has $2x=(5,9,-6,8)-(4,-3,1,7)=(1,12,-7,1),$hence$x=\frac{1}{2}(1,12,-7,1)=\left(\frac{1}{2},6,\frac{-7}{2},\frac{1}{2}\right).$

11. Explain why there does not exist $\lambda\in\mathbb C$ such that$\lambda(2-3i,5+4i,-6+7i)=(12-5i,7+22i,-32-9i).$Solution: If such $\lambda\in\mathbb C$ exists, then we have $\lambda(2-3i)=12-5i\quad\text{and}\quad\lambda(-6+7i)=-32-9i.$It follows that $(2-3i)(-32-9i)=(-6+7i)(12-5i),$this means $-91+78i=-37+114i,$which is impossible. Hence such $\lambda\in\mathbb C$ does not exist.

12. Show that $(x+y)+z=x+(y+z)$ for all $x, y, z\in\mathbb F^n$.

Solution: Suppose $x=(x_1,\cdots,x_n)$, $y=(y_1,\cdots,y_n)$ and $z=(z_1,\cdots,z_n)$. Then \begin{align*} (x+y)+z=&((x_1,\cdots,x_n)+(y_1,\cdots,y_n))+(z_1,\cdots,z_n)\\ =&(x_1+y_1,\cdots,x_n+y_n)+(z_1,\cdots,z_n)\\ =&((x_1+y_1)+z_1,\cdots,(x_n+y_n)+z_n)\\ =&(x_1+(y_1+z_1),\cdots,x_n+(y_n+z_n))\\ =&(x_1,\cdots,x_n)+(y_1+z_1,\cdots,y_n+z_n)\\ =&(x_1,\cdots,x_n)+((y_1,\cdots,y_n)+(z_1,\cdots,z_n))\\ =&x+(y+z). \end{align*}

13. Show that $(ab)x=a(bx)$ for all $x\in\mathbb F^n$ and all $a,b \in\mathbb F$.

Solution: Suppose $x=(x_1,\cdots,x_n)$. Then \begin{align*} (ab)x=&ab(x_1,\cdots,x_n)=((ab)x_1,\cdots,(ab)x_n)\\ =&(a(bx_1),\cdots,a(bx_n))=a(bx_1,\cdots,bx_n)\\ =&a(bx). \end{align*}

14. Show that $1x=x$ for all $x\in\mathbb F^n$.

Solution: Suppose $x=(x_1,\cdots,x_n)$. Then $1x=1(x_1,\cdots,x_n)=(1\cdot x_1,\cdots,1\cdot x_n)=(x_1,\cdots,x_n)=x.$

15. Show that $\lambda(x+y)=\lambda x+\lambda y$ for all $\lambda\in\mathbb F$ and all $\lambda\in\mathbb F^n$.

Solution: Suppose $x=(x_1,\cdots,x_n)$ and $y=(y_1,\cdots,y_n)$. Then \begin{align*} \lambda(x+y)=&\lambda((x_1,\cdots,x_n)+(y_1,\cdots,y_n))\\ =&\lambda(x_1+y_1,\cdots,x_n+y_n)=(\lambda(x_1+y_1),\cdots,\lambda(x_n+y_n))\\ =&(\lambda x_1,\cdots,\lambda x_n)+(\lambda y_1,\cdots,\lambda y_n)\\ =&\lambda x+\lambda y. \end{align*}

16. Show that $(a+b)x=ax+bx$ for all $a,b\in\mathbb F$ and all $x\in\mathbb F^n$.

Solution: Suppose $x=(x_1,\cdots,x_n)$. Then \begin{align*} (a+b)x=&(a+b)(x_1,\cdots,x_n)=((a+b)x_1,\cdots,(a+b)x_n)\\ =&(ax_1+bx_1,\cdots,ax_n+bx_n)\\ =&(ax_1,\cdots,ax_n)+(bx_1,\cdots,bx_n)\\ =&a(x_1,\cdots,x_n)+b(x_1,\cdots,x_n)\\ =&ax+bx. \end{align*}