W214 Linear Algebra Notes - Abstract Vector Spaces

Notes for W214 Linear Algebra

Note for the Student

• Linear algebra is being revisited from a more abstract, mathematical viewpoint.
• Abstraction involves removing extraneous detail to focus on the most important features of a problem, aiding in better understanding and broader applicability.
• Abstract mathematics relies on definitions, theorems, and proofs.
• Active engagement with the material, including writing down definitions and examples, is essential for learning.
• When encountering a worked example (A = B), understanding the meanings of both A and B is crucial before considering the question of their equality.

Chapter 1 Abstract vector spaces

1.1 Introduction

1.1.1 Three different sets

• A set $X$ in mathematics is a collection of distinct objects, called elements of $X$.
• Set $A$ is defined as the set of all ordered pairs $(x, y)$ where $x$ and $y$ are real numbers, represented as $A := {(a<em>1, a</em>2) : a<em>1, a</em>2 ∈ R}$.
• Set $B$ is the set of all ordered real triples $(b<em>1, b</em>2, b<em>3)$ satisfying the equation $b</em>1 − b<em>2 + b</em>3 = 0$, represented as $B := {(b<em>1, b</em>2, b<em>3) : b</em>1, b<em>2, b</em>3 ∈ R \text{ and } b<em>1 − b</em>2 + b_3 = 0}$.
• Set $C$ is the set of all polynomials of degree less than or equal to 4, represented as $C := {\text{polynomials of degree } ≤ 4}$.

1.1.2 Features the sets have in common

• Addition: In each set, there is a natural addition operation where two elements of the set can be added to get a third element.
  • In set $A$, two elements $\mathbf{a} = (a<em>1, a</em>2)$ and $\mathbf{a'} = (a'<em>1, a'</em>2)$ are added as $(a<em>1, a</em>2) + (a'<em>1, a'</em>2) := (a<em>1 + a'</em>1, a<em>2 + a'</em>2)$.
  • In set $B$, two elements $\mathbf{b} = (b<em>1, b</em>2, b<em>3)$ and $\mathbf{b'} = (b'</em>1, b'<em>2, b'</em>3)$ are added as $(b<em>1, b</em>2, b<em>3) + (b'</em>1, b'<em>2, b'</em>3) := (b<em>1 + b'</em>1, b<em>2 + b'</em>2, b<em>3 + b'</em>3)$, ensuring the result satisfies the condition for membership in $B$.
  • In set $C$, two polynomials are added algebraically by adding their corresponding coefficients: $[c<em>4x^4 + c</em>3x^3 + c<em>2x^2 + c</em>1x^1 + c<em>0] + [d</em>4x^4 + d<em>3x^3 + d</em>2x^2 + d<em>1x^1 + d</em>0] := (c<em>4 + d</em>4)x^4 + (c<em>3 + d</em>3)x^3 + (c<em>2 + d</em>2)x^2 + (c<em>1 + d</em>1)x^1 + (c<em>0 + d</em>0)$. Alternitvely, polynomials can be seen as functions.$c(x) + d(x)$.
• Zero element: Each set has a zero element, $\mathbf{0}$, which when added to another element, leaves that element unchanged.
  • In $A$, the zero element is $\mathbf{0} := (0, 0) ∈ A$.
  • In $B$, the zero element is $\mathbf{0} := (0, 0, 0) ∈ B$.
  • In $C$, the zero element is the zero polynomial $0 = 0x^4 + 0x^3 + 0x^2 + 0x + 0$.
• Multiplication by scalars: In each set, elements can be multiplied by any real number, resulting in another element within the set.
  • In $A$, if $\mathbf{a} = (a<em>1, a</em>2)$, then $k.\mathbf{a} = (ka<em>1, ka</em>2)$, where $k ∈ R$.
  • In $B$, $k(u<em>1, u</em>2, u<em>3) := (ku</em>1, ku<em>2, ku</em>3)$.
  • In $C$, scalar multiplication is defined as $k.[c<em>4x^4 +c</em>3x^3 +c<em>2x^2 +c</em>1x+c<em>0] = kc</em>4x^4 +kc<em>3x^3 +kc</em>2x^2 +kc<em>1x+kc</em>0$.

1.1.3 Features that the sets do not have

• Set $A = R^2$ has a multiplication operation, Set $B$ and $C$ do not.
• Set $C$ has a “take the derivative” operation, $c \rightarrow \frac{d}{dx}c$.

1.1.4 Rules

• Addition operation in sets $A$, $B$, and $C$ satisfies rules like commutativity, $a + a' = a' + a$ and associativity $(x + y) + z = x + (y + z)$.

1.2 Definition of an abstract vector space

A vector space is a set $V$ equipped with:

• D1: An addition operation $u, v ∈ V \rightarrow u + v ∈ V$.
• D2: A zero vector $0 ∈ V$.
• D3: A scalar multiplication operation $k ∈ R, v ∈ V \rightarrow k.v ∈ V$.

This data should satisfy the following rules for all $u, v, w$ belongning to $V$ and for all real numbers $k$ and $l$:

• R1: $v + w = w + v$
• R2: $(u + v) + w = u + (v + w)$
• R3a: $0 + v = v$
• R3b: $v + 0 = v$
• R4: $k.(v + w) = k.v + k.w$
• R5: $(k + l).v = k.v + l.v$
• R6: $k.(l.v) = (kl).v$
• R7: $1.v = v$
• R8: $0.v = 0$

1.3 First example of a vector space

• The set $B$ can be shown to be a vector space by defining addition, zero vector, and scalar multiplication and verifying that the conditions R1 – R8 are met.
  • D1. Addition: define $\mathbf{u} + \mathbf{v}$.
  • D2. Zero vector: define $\mathbf{0} ∈ B$.
  • D3. Scalar Multiplication: define $k.\mathbf{u}$.
  • Rules: Check for each rule ($R1$ – $R8$).

1.4 More examples and non-examples

• The set $Z := {z}$ is a vector space with addition defined as $z + z := z$, zero element defined as $0 := z$, and scalar multiplication defined as $k.z := z$.
• The set $R^n := {(x<em>1, x</em>2, …, x<em>n) : x</em>i ∈ R \text{ for all } i = 1 . . . n}$ is a vector space with componentwise addition and scalar multiplication.
• The set $R^∞ := {(x1, x2, x3, . . .) : xi ∈ R \text{ for all } i = 1, 2, 3, . . .}$ is a vector space under componentwise addition and scalar multiplication.
• The set $Fun(X) := {f : X → R}$ of real-valued functions on a set $X$ is a vector space when equipped with addition defined as $(f + g)(x) := f(x) + g(x), x ∈ X$ and scalar multiplication defined as $(k.f)(x) := kf(x)$.
• The set $Mat_{n,m}$ of all $n × m$ matrices is a vector space.

1.5 Some results about abstract vector spaces

• Lemma 1.5.1: If $V$ is a vector space with zero vector $\mathbf{0}$, and $0'$ is a vector in $V$ satisfying $0' + v = v$ for all $v ∈ V$, then $0' = 0$.
• Definition 1.5.2: If $V$ is a vector space, the additive inverse of a vector $v ∈ V$ is defined as $-v := (−1).v$
• Lemma 1.5.3: If $V$ is a vector space, then for all $v ∈ V$ we have $-v + v = 0$ and $v + (−v) = 0$.
• Lemma 1.5.4: Suppose that two vectors $w$ and $v$ in a vector space satisfy $w + v = 0$. Then $w = −v$.
• Lemma 1.5.5: Let $V$ be a vector space and $k$ any scalar. Then $k.0 = 0$.
• Lemma 1.5.6: Suppose that $v$ is a vector in a vector space $V$ and that $k$ is a scalar. Then $k.v = 0 ⇔ k = 0 \text{ or } v = 0$.

1.6 Subspaces

• A subset $U ⊆ V$ of a vector space $V$ is a subspace if:

  • For all $\mathbf{u}, \mathbf{u'} ∈ U, \mathbf{u} + \mathbf{u'} ∈ U$
  • $\mathbf{0} ∈ U$
  • For all scalars $k$ and all vectors $\mathbf{u} ∈ U, k.\mathbf{u} ∈ U$

• If $U$ is a subspace of a vector space $V$, then with the restricted operations addition and scalar multiplication from $V$, $U$ is also a vector space.

• Examples of subspaces:

  • A line $L$ through the origin in $R^2$.
  • Lines and planes in $R^3$.
  • The set $\mathbb{{0}}$.
  • Hyperplanes orthogonal to a fixed vector.
  • Continuous functions $Cont(I)$.
  • Differentiable functions $Diff(I)$.
  • Vector spaces of polynomials $Poly$ and $Poly_n$.
  • Polynomials in many variables $Poly[x, y] and $Poly_n[x, y].
  • Polynomial vector fields $Vect_n(R^2).
  • Trigonometric polynomials $Trig and $Trig_n.
  • Solutions to homogenous linear differential equations.