Math 223, Section 201

Homework

There will be 11 homework assignments. Homework is due at 10:00,
before class begins. No late homework will be accepted.

Your two lowest homework scores will be dropped. No further concession
will be made, for any (including medical) reason.


Homework 1. Due Wednesday, January 15.  

Writing.  Problems 1a, 1b, 1c, 1f, 1g, 2, 3 from Section 8, Chapter 2 of Curtis.
For Problems 1, find the general solution in parametric vector form.
Also, do the following 2 problems from previous Math 221 exams:



Homework 2. Due Wednesday, January 22.  

Reading.  Jänich 1.1, 1.2, 1.3.

Writing.  Problems 1.1, 1.2, 1.3 from Jänich, plus the following:

Let \(f:X\to Y\) be a map, $A,A'\subset X$ subsets of $X$, and $B,B'\subset Y$ subsets of $Y$.
Prove that $f(A\cup A')=f(A)\cup f(A')$, and that $f^{-1}(B\cup B')=f^{-1}(B)\cup f^{-1}(B')$.
Homework 3. Due Wednesday, January 29.  

Reading.  Jänich 2.1, 2.2, 2.3, 2.4, 2.5.

Writing.  From Jänich:
2.1 (only for real vector spaces),
2.2, 2.3, plus the following:

(A) Find all complex numbers $z$, such that $z^3=1$. In other words,
find all $(a,b)\in\mathbb{C}$, such that $(a,b)(a,b)(a,b)=(1,0)$.

(B) Let $\mathbb{F}=\{x\in\mathbb{R}\mid\exists a,b\in\mathbb{Q}\colon x=a+b\sqrt{5}\}$.
Prove that $x,y\in\mathbb{F}\Rightarrow x+y\in\mathbb{F}$, and $x,y\in\mathbb{F}\Rightarrow x\cdot y\in\mathbb{F}$.
Prove that $\mathbb{F}$, with addition and multiplication induced from $\mathbb{R}$
(these are well-defined operations on $\mathbb{F}$ by what you just proved)
is a field.
Prove that for $a,b,c,d\in\mathbb{Q}$, we have $a+b\sqrt{5}=c+d\sqrt{5}\Rightarrow \text{$a=c$ and $b=d$}$.
The field $\mathbb{F}$ is usually denoted by $\mathbb{F}=\mathbb{Q}(\sqrt{5})$.
Homework 4. Due Wednesday, February 12.  

Reading.  Jänich: Sections 3.1, 3.2, 3.3, 3.4.
Curtis: Exercises 4.1, 4.3, 4.4, 4.5 (Do not hand in).

Writing.  From Jänich:
3.1, 3.2, 3.3, $\ast$, plus the following:

(A) If $U_1$ and $U_2$ are complementary subspaces of $V$ (see Problem 3.2),
then we write $V=U_1\oplus U_2$.
Prove that if $V=U_1\oplus U_2$, then every vector $v\in V$ can be written uniquely
as $v=u_1+ u_2$, with $u_1\in U_1$ and $u_2\in U_2$.
Homework 5. Due Wednesday, February 26.  

Reading.  Jänich: Sections 4.1, 4.2, 4.3, 5.1, 5.2, 5.3, 5.4, 5.5.

Writing.  From Jänich:
4.1, 4.2, 5.1, 5.2, 5.3.

plus the following:

(A) Prove that
$\{A\in M(3\times3,\mathbb{R})\mid \text{$A$ commutes with $\scriptstyle\begin{pmatrix}2&0&0\\0&3&1\\0&0&3\end{pmatrix}$}\}$
is a subspace of $M(3\times3,\mathbb{R})$, find a basis, and determine its dimension.

(B) Consider the maps $D,I,P:C^\infty(\mathbb{R},\mathbb{R})\to C^\infty(\mathbb{R},\mathbb{R})$, defined by $D(f)=f'$, $(If)(x)=\int_0^xf(t)\,dt$, $(Pf)(x)=xf(x)$. Prove that $D,I,P$ are linear endomorphisms, by quoting suitable statements from Calculus. Find $DI$, $ID$, $DP-PD$. Find kernel and image of $ID$ and $P$.

(C) Find the standard matrix of the reflection across the line spanned by $(2,5)$ in $\mathbb{R}^2$.
Homework 6. Due Friday, March 6.  

Reading.  Jänich: Sections 6.1, 6.2, 6.3,
6.4: know the statements of the definition and the theorem, and the formula at the top of Page 112,
6.5, 6.6, 6.7.

Writing.  Jänich:   6.1, 6.2, 6.3 ($E$ is the identity matrix).
Curtis:   19.4, 19.7, 19.10.

Homework 7. Due Friday, March 27.  

Reading.  Jänich: Sections 8.1, 8.2, 9.1, 9.2, 9.3, 9.4.

Writing.  Jänich:   8.1, 8.2 ($a_{2,3}=1$), 8.2*, 8.2P, 9.1, 9.2, 9.3.

Homework 8. Due Friday, April 3.  

Reading.  Jänich: Sections 8.3, 8.4, 8.5, 10.1, 10.2, 10.3, 10.4, 10.5.

Writing.  Jänich:   8.3, 9.1*, 9.2P, 9.3P, 10.1, 10.3.

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