Vector Spaces

1A. $\mathbb{R}^n$ and $\mathbb{C}^n$

1) Complex Numbers

Complex Number

A complex number is an ordered pair $(a,b)$ of real numbers, written as

$$a+bi,$$

where

$$i^2=-1.$$

Meaning: A complex number consists of a real part and an imaginary part.

Complex Numbers ($\mathbb{C}$)

The set of all complex numbers is

$$\mathbb{C} = \{a+bi\mid a,b\in\mathbb{R}\}.$$

Meaning: $\mathbb{C}$ contains every complex number.

Real Numbers as Complex Numbers

A real number $a$ is identified with $a+0i$.

Thus,

$$\mathbb{R}\subseteq\mathbb{C}.$$

Meaning: Every real number is also a complex number.

Complex Addition

For $a,b,c,d\in\mathbb{R}$,

$$(a+bi)+(c+di) = (a+c)+(b+d)i.$$

Meaning: Add the real parts and imaginary parts separately.

Complex Multiplication

For $a,b,c,d\in\mathbb{R}$,

$$(a+bi)(c+di) = (ac-bd)+(ad+bc)i.$$

Meaning: Multiply normally and use $i^2=-1$.

2) Properties of Complex Arithmetic

Commutativity

For $\alpha,\beta\in\mathbb{C}$,

$$\alpha+\beta=\beta+\alpha,$$ $$\alpha\beta=\beta\alpha.$$

Meaning: Changing the order does not change the result.

Associativity

For $\alpha,\beta,\lambda\in\mathbb{C}$,

$$(\alpha+\beta)+\lambda = \alpha+(\beta+\lambda),$$ $$(\alpha\beta)\lambda = \alpha(\beta\lambda).$$

Meaning: Changing the grouping does not change the result.

Additive Identity

For every $\lambda\in\mathbb{C}$,

$$\lambda+0=\lambda.$$

Meaning: Adding zero changes nothing.

Multiplicative Identity

For every $\lambda\in\mathbb{C}$,

$$\lambda1=\lambda.$$

Meaning: Multiplying by one changes nothing.

Additive Inverse

For every $\alpha\in\mathbb{C}$, there is a unique $-\alpha$ such that

$$\alpha+(-\alpha)=0.$$

Meaning: Every complex number has an opposite that cancels it.

Multiplicative Inverse

For every nonzero $\alpha\in\mathbb{C}$, there is a unique $1/\alpha$ such that

$$\alpha\left(\frac{1}{\alpha}\right)=1.$$

Meaning: Every nonzero complex number has a reciprocal.

Distributive Property

For $\lambda,\alpha,\beta\in\mathbb{C}$,

$$\lambda(\alpha+\beta) = \lambda\alpha+\lambda\beta.$$

Meaning: Multiplication distributes over addition.

3) Subtraction and Division

Subtraction

For $\alpha,\beta\in\mathbb{C}$,

$$\beta-\alpha = \beta+(-\alpha).$$

Meaning: Subtraction means adding the additive inverse.

Division

For $\alpha\neq0$,

$$\frac{\beta}{\alpha} = \beta\left(\frac{1}{\alpha}\right).$$

Meaning: Division means multiplying by the reciprocal.

4) Scalars and $\mathbb{F}$

$\mathbb{F}$

Throughout the chapter,

$$\mathbb{F} = \mathbb{R} \quad\text{or}\quad \mathbb{C}.$$

Meaning: Results involving $\mathbb{F}$ apply to both real and complex numbers.

Scalar

An element of $\mathbb{F}$ is called a scalar.

Meaning: A scalar is a real or complex number used to multiply vectors.

5) Lists

List

A list of length $n$ is an ordered collection of $n$ elements.

Meaning: In a list, both the elements and their order matter.

List Equality

Two lists are equal if they have the same length and the same elements in the same order.

Meaning: Changing the order or an entry produces a different list.

6) $\mathbb{F}^n$

$\mathbb{F}^n$

$\mathbb{F}^n$ is the set of all lists of length $n$ whose entries belong to $\mathbb{F}$.

$$\mathbb{F}^n = \{(x_1,\ldots,x_n)\mid x_k\in\mathbb{F}\}.$$

Meaning: An element of $\mathbb{F}^n$ is an ordered list of $n$ real or complex numbers.

Coordinate

For

$$x=(x_1,\ldots,x_n)\in\mathbb{F}^n,$$

$x_k$ is called the $k$th coordinate of $x$.

Meaning: A coordinate is one entry of a vector.

7) Addition in $\mathbb{F}^n$

Vector Addition

For $x,y\in\mathbb{F}^n$,

$$(x_1,\ldots,x_n)+(y_1,\ldots,y_n) = (x_1+y_1,\ldots,x_n+y_n).$$

Meaning: Vectors are added coordinate by coordinate.

Zero Vector

The zero vector is

$$0=(0,\ldots,0).$$

Meaning: Every coordinate of the zero vector is zero.

Additive Inverse

If

$$x=(x_1,\ldots,x_n),$$

then

$$-x=(-x_1,\ldots,-x_n).$$

Thus,

$$x+(-x)=0.$$

Meaning: Negating every coordinate gives the vector that cancels $x$.

8) Scalar Multiplication in $\mathbb{F}^n$

Scalar Multiplication

For $\lambda\in\mathbb{F}$,

$$\lambda(x_1,\ldots,x_n) = (\lambda x_1,\ldots,\lambda x_n).$$

Meaning: A scalar multiplies every coordinate of a vector by the same value.

1B. Definition of Vector Space

1) Vector Operations

Vector Addition

An addition on $V$ assigns a vector

$$u+v\in V$$

to every $u,v\in V$.

Meaning: Adding two vectors in $V$ must produce another vector in $V$.

Scalar Multiplication

A scalar multiplication on $V$ assigns a vector

$$\lambda v\in V$$

to every $\lambda\in\mathbb{F}$ and $v\in V$.

Meaning: Multiplying a vector by a scalar must produce another vector in $V$.

2) Vector Space

Vector Space

A vector space over $\mathbb{F}$ is a set $V$ with vector addition and scalar multiplication satisfying the vector space properties.

Meaning: A vector space is a set whose elements behave consistently under addition and scalar multiplication.

3) Vector Space Properties

Commutativity of Addition

For all $u,v\in V$,

$$u+v=v+u.$$

Meaning: The order of vectors does not matter when adding them.

Associativity of Addition

For all $u,v,w\in V$,

$$(u+v)+w = u+(v+w).$$

Meaning: The grouping of vectors does not matter when adding them.

Additive Identity

There exists $0\in V$ such that

$$v+0=v$$

for every $v\in V$.

Meaning: Every vector space contains a zero vector.

Additive Inverse

For every $v\in V$, there exists $-v\in V$ such that

$$v+(-v)=0.$$

Meaning: Every vector has an opposite vector that cancels it.

Associativity of Scalar Multiplication

For all $\alpha,\beta\in\mathbb{F}$ and $v\in V$,

$$(\alpha\beta)v = \alpha(\beta v).$$

Meaning: Scalars can be multiplied before or after they are applied to a vector.

Multiplicative Identity

For every $v\in V$,

$$1v=v.$$

Meaning: Multiplying a vector by one leaves it unchanged.

Distributivity over Vector Addition

For $\lambda\in\mathbb{F}$ and $u,v\in V$,

$$\lambda(u+v) = \lambda u+\lambda v.$$

Meaning: A scalar distributes over vector addition.

Distributivity over Scalar Addition

For $\alpha,\beta\in\mathbb{F}$ and $v\in V$,

$$(\alpha+\beta)v = \alpha v+\beta v.$$

Meaning: Scalar addition distributes over scalar multiplication.

4) Vectors and Points

Vector

An element of a vector space is called a vector or a point.

Meaning: A vector does not have to be an arrow or coordinate list; it can be any object belonging to a vector space.

5) Real and Complex Vector Spaces

Real Vector Space

A vector space over $\mathbb{R}$ is called a real vector space.

Meaning: Its scalars are real numbers.

Complex Vector Space

A vector space over $\mathbb{C}$ is called a complex vector space.

Meaning: Its scalars are complex numbers.

6) Function Spaces

$\mathbb{F}^S$

For a set $S$, $\mathbb{F}^S$ denotes the set of all functions from $S$ to $\mathbb{F}$.

$$f:S\rightarrow\mathbb{F}.$$

Meaning: Functions themselves can be treated as vectors.

Function Addition

For $f,g\in\mathbb{F}^S$,

$$(f+g)(x) = f(x)+g(x).$$

Meaning: Functions are added by adding their values at each input.

Scalar Multiplication of Functions

For $\lambda\in\mathbb{F}$ and $f\in\mathbb{F}^S$,

$$(\lambda f)(x) = \lambda f(x).$$

Meaning: Scalar multiplication multiplies every function value by the scalar.

7) Elementary Properties

Unique Additive Identity

A vector space has exactly one additive identity.

Meaning: A vector space has only one zero vector.

Unique Additive Inverse

Every vector has exactly one additive inverse.

Meaning: Each vector has exactly one vector that cancels it.

Scalar Zero Times a Vector

For every $v\in V$,

$$0v=0.$$

Meaning: Multiplying any vector by the scalar zero gives the zero vector.

Scalar Times the Zero Vector

For every $\lambda\in\mathbb{F}$,

$$\lambda0=0.$$

Meaning: Multiplying the zero vector by any scalar still gives the zero vector.

Multiplication by $-1$

For every $v\in V$,

$$(-1)v=-v.$$

Meaning: Multiplying a vector by $-1$ gives its additive inverse.

1C. Subspaces

1) Subspace

Subspace

A subset $U$ of a vector space $V$ is a subspace if $U$ is itself a vector space using the same addition and scalar multiplication as $V$.

Meaning: A subspace is a smaller vector space contained inside another vector space.

2) Conditions for a Subspace

Subspace Test

A subset $U\subseteq V$ is a subspace if and only if the following three conditions hold.

Additive Identity

$$0\in U.$$

Meaning: A subspace must contain the zero vector.

Closed under Addition

If $u,w\in U$, then

$$u+w\in U.$$

Meaning: Adding vectors in the subspace cannot take us outside the subspace.

Closed under Scalar Multiplication

If $\lambda\in\mathbb{F}$ and $u\in U$, then

$$\lambda u\in U.$$

Meaning: Multiplying a vector by a scalar cannot take us outside the subspace.

3) Sum of Subspaces

Sum of Subspaces

If $V_1,\ldots,V_m$ are subspaces of $V$, their sum is

$$V_1+\cdots+V_m = \{v_1+\cdots+v_m \mid v_k\in V_k\}.$$

Meaning: The sum contains every vector obtained by adding one vector from each subspace.

Smallest Containing Subspace

$V_1+\cdots+V_m$ is the smallest subspace of $V$ containing all of

$$V_1,\ldots,V_m.$$

Meaning: Any subspace containing all the $V_k$ must also contain their sum.

4) Direct Sum

Direct Sum

The sum

$$V_1+\cdots+V_m$$

is called a direct sum if every vector in the sum has exactly one representation

$$v_1+\cdots+v_m,$$

where

$$v_k\in V_k.$$

It is written as

$$V_1\oplus\cdots\oplus V_m.$$

Meaning: Each vector can be uniquely separated into components from the subspaces.

5) Condition for a Direct Sum

Zero-Representation Condition

The sum

$$V_1+\cdots+V_m$$

is direct if and only if

$$v_1+\cdots+v_m=0$$

implies

$$v_1=\cdots=v_m=0.$$

Meaning: The zero vector must have only the trivial decomposition.

6) Direct Sum of Two Subspaces

Two-Subspace Direct Sum Test

For subspaces $U$ and $W$,

$$U+W\text{ is a direct sum} \iff U\cap W=\{0\}.$$

Meaning: Two subspaces form a direct sum exactly when their only common vector is the zero vector.