Vector Spaces

[Notes from Linear Algebra Done Right | Chapter 1]

𝐅 stands for either 𝐑 or 𝐂.

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

• $\mathbb{R}$ → real numbers
• $\mathbb{C}$ → complex numbers

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Scalars

A scalar is a single number.

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

In

$$x=(2,-3,5)\in {\mathbb{R}}^3$$

then $2,-3,5$ are the three scalar coordinates of $x$.

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Lists : A list of length 𝑛 is an ordered collection of 𝑛 elements. Two lists are equal if and only if they have the same length and the same elements in the same order.

Every list has a finite, nonnegative integer length.

A list of length $n$ can be written as:

$$(z_1,\dots ,z_n)$$

Another name for a list of length $n$ is an $n$-tuple.

Lists care about order and repetition, sets do not.

For example:

$$(3,5)\ne (5,3)$$

because the order is different.

Also:

$$(4,4)\ne (4,4,4)$$

We already know:

$${\mathbb{R}}^2=\{(x,y):x,y\in \mathbb{R}\}$$

and

$${\mathbb{R}}^3=\{(x,y,z):x,y,z\in \mathbb{R}\}$$

To generalize this to any dimension, we need the idea of a list.

${\mathbb{F}}^n$

${\mathbb{F}}^n$ is the set of all lists of length $n$ whose elements are scalars from $\mathbb{F}$.

$$\boxed{{\mathbb{F}}^n=\text{all lists of length }n\text{ whose elements are in }\mathbb{F}}$$ $$\boxed{{\mathbb{F}}^n(x_1,\dots ,x_n):x_k\in \mathbb{F}\text{ for }k=1,\dots ,n}$$

So if:

$$x\in {\mathbb{R}}^5$$

then $x$ is simply a list of 5 real-number scalars:

$$x=(x_1,x_2,x_3,x_4,x_5)$$

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Vectors

$$x=(x_1,x_2,\dots ,x_n)\in {\mathbb{F}}^n$$

means that $x$ is an ordered list of $n$ scalars.

Example:

$$x=(2,-1,5)\in {\mathbb{R}}^3$$

where:

$$x_1=2,x_2=-1,x_3=5$$ $$\boxed{{\mathbb{R}}^n=\text{all vectors with }n\text{ real components}}$$

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Vector Addition

Vectors are added component-wise:

$$(x_1,\dots ,x_n)+(y_1,\dots ,y_n)=(x_1+y_1,\dots ,x_n+y_n)$$

The vectors must belong to the same ${\mathbb{F}}^n$.

So:

$$(1,2,3)+(4,5)$$

is not defined.

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Scalar Multiplication

For a scalar $\lambda \in \mathbb{F}$:

$$\lambda (x_1,\dots ,x_n)=(\lambda x_1,\dots ,\lambda x_n)$$

Scalar zero

$$0\in \mathbb{F}$$

Zero vector

$$0=(0,\dots ,0)\in {\mathbb{F}}^n$$

This is the zero vector in ${\mathbb{F}}^n$.

It has $n$ coordinates, all equal to the scalar $0$.

For example:

$$x+0=x$$

Here the $0$ must be the zero vector, because adding a vector and a scalar is not defined.

Additive Inverse

For:

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

we define:

$$-x=(-x_1,\dots ,-x_n)$$

such that:

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

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A field is a set containing at least two distinct elements called 0 and 1, along with operations of addition and multiplication while satisfying all properties commutativity, associativity, identities, inverses, and distributivity.

𝐑 and 𝐂 are fields

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Vector Space

FORMALLY A vector space $V$ over the field $\mathbb{R}$ is a set of vectors together with two operations:

• Vector addition : $+:V\times V\to V$ Add two vectors to get another vector.
• Scalar multiplication $\cdot :\mathbb{R}\times V\to V$ Multiply a vector by a real number (scalar).

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An addition on $V$ is an operation that takes any two vectors $u,v\in V$ and produces another vector in $V$:

$$u+v\in V$$

So:

$$V\times V\to V$$

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Scalar Multiplication

A scalar multiplication on $V$ takes a scalar $\lambda \in \mathbb{F}$ and a vector $v\in V$ and produces another vector in $V$:

$$\lambda v\in V$$

So:

$$\mathbb{F}\times V\to V$$

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A vector space is a set $V$ where we can:

1. add two vectors
2. multiply a vector by a scalar and these operations obey a specific set of rules called the vector space axioms

These are the vector-space axioms.

Property	Rule
Commutativity	$u+v=v+u$
Associativity	$(u+v)+w=u+(v+w)$
Scalar associativity	$(ab)v=a(bv)$
Additive identity	$v+0=v$
Additive inverse	$v+(-v)=0$
Multiplicative identity	$1v=v$
Distributivity	$a(u+v)=au+av$
Distributivity	$(a+b)v=av+bv$

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Examples of Vector Spaces

The simplest possible vector space is:

$$\boxed{\{0\}}$$

It contains exactly one vector: the zero vector.

It still satisfies all the vector-space axioms.

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${\mathbb{F}}^{\infty }$ — Infinite Sequences

${\mathbb{F}}^{\infty }$ is the set of all infinite sequences of elements of $\mathbb{F}$:

$${\mathbb{F}}^{\infty }(x_1,x_2,\dots ):x_k\in \mathbb{F}\text{ for }k=1,2,\dots$$

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${\mathbb{F}}^S$ — Functions as Vectors

If $S$ is a set, then:

$$\boxed{{\mathbb{F}}^S=\text{all functions }f:S\to \mathbb{F}}$$

So ${\mathbb{F}}^S$ is the set of all $\mathbb{F}$-valued functions defined on $S$. It lets linear algebra work with functions, not just numerical lists.

The functions themselves are the vectors.

Example: ${\mathbb{R}}^{[0,1]}$

$$f(x)=x^2$$ $$g(x)=\sin (x)$$ $$(f+g)(x)=x^2+\sin (x)$$

The elements of the vector space ${\mathbb{R}}^{[0,1]}$ are real-valued functions on $[0,1]$, not lists.

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In ${\mathbb{F}}^n$:

$$(x_1,\dots ,x_n)$$

is a vector.

But in ${\mathbb{F}}^S$, a function is a vector.

For example:

$$f(x)=x^2$$

can be a vector.

$${\mathbb{F}}^S$$

as:

all possible ways of assigning a scalar from $\mathbb{F}$ to every element of $S$.

Then:

• $S=1,\dots ,n$ → ${\mathbb{F}}^n$
• $S=1,2,\dots$ → ${\mathbb{F}}^{\infty }$
• $S=[0,1]$ → functions on $[0,1]$

In general, a vector space is an abstract structure whose elements might be lists, functions, matrices, polynomials, or other mathematical objects.

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Elements of a vector space are called vectors or points.

A vector does not have to look like $(x_1,x_2,x_3)$. Depending on the vector space, a vector could be:

• A column of numbers: $\left(\begin{array}{c}1\\ 2\\ 3\end{array}\right)$
• A matrix: $\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)$
• A polynomial: $1+2x+3x^2$
• A function:$f(x)=\sin x$

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Uniqueness of the additive identity

• A vector space has exactly one zero vector.

Uniqueness of the additive inverse

• Every vector has exactly one additive inverse.

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

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Subspaces

A subspace is a subset of a vector space that is itself a vector space, using the same:

• zero vector
• addition
• scalar multiplication

as the original vector space.

If:

$$U\subseteq V$$

then $U$ is a subspace of $V$ if $U$ is itself a vector space under the operations inherited from $V$.

A subset $U\subseteq V$ is a subspace iff these three conditions hold:

• Contains the zero vector

$$\boxed{0\in U}$$

• Closed under addition

For every $u,w\in U$:

$$\boxed{u+w\in U}$$

• Closed under scalar multiplication

For every $a\in \mathbb{F}$ and $u\in U$:

$$\boxed{au\in U}$$

If all three are true:

$$\boxed{U\text{ is a subspace of }V}$$

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Sums of Subspaces

The union of two subspaces is generally not a subspace.

Instead, linear algebra uses the sum of subspaces.

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

$$\boxed{V_1+\cdots +V_m=v_1+\cdots +v_m:v_1\in V_1,\dots ,v_m\in V_m}$$

Example in ${\mathbb{F}}^3$

Let:

$$U=(x,0,0):x\in \mathbb{F}$$

and:

$$W=(0,y,0):y\in \mathbb{F}$$

Then:

$$(x,0,0)+(0,y,0)=(x,y,0)$$

Therefore:

$$\boxed{U+W=(x,y,0):x,y\in \mathbb{F}}$$

So combining the two subspaces gives the entire $xy$-plane inside ${\mathbb{F}}^3$.

Example in ${\mathbb{F}}^4$

$$U=(x,x,y,y):x,y\in \mathbb{F}$$ $$W=(x,x,x,y):x,y\in \mathbb{F}$$ $$u=(a,a,b,b)\in U$$ $$w=(c,c,c,d)\in W$$ $$u+w=(a+c,a+c,b+c,b+d)$$ $$U+W\subseteq (x,x,y,z):x,y,z\in \mathbb{F}$$

To show equality, every vector of the form $(x,x,y,z)$ must be expressible as $u+w$.

Indeed:

$$(x,x,y,z)=(x,x,y,y)+(0,0,0,z-y)$$

where:

$$(x,x,y,y)\in U$$ $$(0,0,0,z-y)\in W$$ $$\boxed{U+W=(x,x,y,z):x,y,z\in \mathbb{F}}$$

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The sum of subspaces is itself a subspace:

$$\boxed{V_1+\cdots +V_m\text{ is a subspace of }V}$$ $$\boxed{V_1+\cdots +V_m=\text{smallest subspace containing }{V}_1,\dots ,V_m}$$

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Direct Sums

Suppose $V_1,\dots ,V_m$ are subspaces of $V$.

Every vector in their sum can be written as

$$v=v_1+\cdots +v_m,v_k\in V_k.$$

                v = (3,5,7)
                     ↓
         ┌───────────┼───────────┐
         ↓           ↓           ↓
   v₁ = (3,0,0) v₂ = (0,5,0) v₃ = (0,0,7)
         V₁          V₂          V₃
      x-axis       y-axis       z-axis

$$\boxed{V_1+\cdots +V_m\text{ is a direct sum}}$$

if every vector in $V_1+\cdots +V_m$ has exactly one representation

$$v=v_1+\cdots +v_m,v_k\in V_k.$$

We write:

$$\boxed{V_1\oplus \cdots \oplus V_m}$$

The $\oplus$ means: the sum is direct / representations are unique.

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Example: direct sum in ${\mathbb{F}}^3$

$$U=(x,y,0):x,y\in \mathbb{F}$$ $$W=(0,0,z):z\in \mathbb{F}.$$

Every vector $(x,y,z)$ can be written as

$$(x,y,z)=(x,y,0)+(0,0,z).$$ $${\mathbb{F}}^3=U+W.$$

This representation is unique.

Therefore:

$$\boxed{{\mathbb{F}}^3=U\oplus W}$$

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How to test a direct sum

A sum is direct iff the zero vector has only the trivial representation.

$$\boxed{V_1+\cdots +V_m\text{ is direct}⟺v_1+\cdots +v_m=0\Rightarrow v_1=\cdots =v_m=0}$$

where $v_k\in V_k$.

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Direct sum of two subspaces

For two subspaces there is an especially useful test:

$$\boxed{U+W\text{ is direct}⟺U\cap W=0}$$

Note

Pairwise trivial intersections are sufficient for a direct sum of two subspaces, but NOT sufficient when there are three or more subspaces.

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