Chapter 1 · Kartikay Dubey's Blog

Complex Numbers

Complex number is an ordered pair $(𝑎, 𝑏)$ where $𝑎, 𝑏 \in \mathbb{R}$ , but we will write this as $a+b\iota$ . where $\iota = \sqrt{-1}$ Note:

\mathbb{R} \subset \mathbb{C}

Properties of complex numbers:

Commutativity
Associativity
multiplicative identity = 1
additive identity = 0
additive inverse exists
multiplicative inverse exists (except 0)
distributive property

Lists

We define a list as:

L = (a_1,a_2 ... a_n)\ , n \ne 0

L_1 = L_2 \iff n(L_1) = n(L_2)\ \& \ a_i = b_i, \forall i, a_i \in L_1, b_i \in L_2;

$\mathbb{F}^n$ is the set of all lists of length 𝑛 of elements of 𝐅:

Fields

A field is a set containing at least two distinct elements called 0 and 1, along with operations of addition and multiplication satisfying all properties listed for Complex numbers above.

Algebraically closed field

every non-constant polynomial with coefficients in 𝐅 has a zero

Vector space

A vector space over a field F is a set V with:

+:V\times V\to V

and

\cdot:\mathbb F\times V\to V

satisfying the usual rules of addition and scalar multiplication.

u+v\in V,\qquad \lambda v\in V

for all

u,v\in V,\qquad \lambda\in\mathbb F.

\lambda u\in\mathbb V ,\ \text{generally}\ uv \text{ is not defined for }u,v\in V.

A vector space is a set 𝑉 along with an addition on 𝑉 and a scalar multiplication on 𝑉 such that the following properties hold.

Commutativity: 𝑢 + 𝑣 = 𝑣 + 𝑢
Associativity: (𝑢 + 𝑣) + 𝑤 = 𝑢 + (𝑣 + 𝑤) & (𝑎𝑏)𝑣 = 𝑎(𝑏𝑣)
multiplicative identity: 1𝑣 = 𝑣
additive identity: element 0 ∈ 𝑉 such that 𝑣 + 0 = 𝑣 for all 𝑣 ∈ 𝑉.
additive inverse exists: there exists 𝑤 ∈ 𝑉 such that 𝑣 + 𝑤 = 0.
distributive property : 𝑎(𝑢 + 𝑣) = 𝑎𝑢 + 𝑎𝑣

Subspaces

$U \subset {V}$ is is called a subspace of $V$ if $U$ is also a vector space with the same additive identity, addition, and scalar multiplication as on $V$ $U$ is a subspace of $V$ iff,

0 \in U

u,w \in U \implies u+w \in U

a \in F, u \in U \implies au \in U

Sum of Subspaces

Let $V_{1},V_{2}\dots V_{m}$ be subspaces of $V$ . then,

V_{1}+V_{2} \dots V_{m} = \{ v_{1}+v_{2}\dots v_{m} : v_{1}\in V_{1},\dots v_{m}\in V_{m} \}

sum of subspaces is always a subspace Suppose $V_{1},V_{2}\dots V_{m}$ are subspaces of $V$ . Then $V_{1}+V_{2}\dots + V_{m}$ is the smallest subspace of $V$ containing $V_{1},V_{2}\dots V_{m}$

Direct Sums

sum $V_{1}+V_{2}\dots + V_{m}$ is called a direct sum if each element of $V_{1}+V_{2}\dots + V_{m}$ can be written in only one way as a sum $v_{1}+v_{2}\dots + v_{m}$ , in this case $V_{1}+V_{2}\dots + V_{m}$ can be written as $V_{1}\oplus V_{2}\dots \oplus V_{m}$

Condition for a direct sum

Suppose $V_{1}+V_{2}\dots + V_{m}$ are subspaces of $V$ . Then $V_{1}+V_{2}\dots + V_{m}$ is a direct sum if and only if the only way to write $0$ as a sum $v_{1}+v_{2}\dots+v_{m}$ , where each $v_{k} \in V_{k}$ , is by taking each $v_{k}$ equal to $0$

Direct sum of two subspaces

Suppose $U$ and $W$ are subspaces of $V$ , then $U+W$ is a direct sum $\iff$ $U\cap W=\{ 0 \}$