Vector Spaces and Subspaces

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Introduction

MATH1012 Chapter 2 Goals

• Determine when a set of vectors is a subspace
• Determine when a set of vectors is linearly independent
• Find a basis for a subspace, and hence determine its dimension

Vectors

• A (real) vector is an ordered $n$-tuple of real numbers
  • A vector is the coordinates of points in $n$-dimensional space
• The set of all vectors of arity $n$ is denoted ${\mathbb{R}}_n$

${\mathbb{R}}_2$ is the standard $xy$ plane, ${\mathbb{R}}_3$ is 3D space
But we can go beyond that into spaces we can’t visualise geometrically

• Vectors can only be added with vectors with the same arity
  • e.g. you cannot multiply a 2D and 3D vector

Vectors Y11
Vectors Y12

Sets of Vectors

• We frequently need to describe / analyse various sets of vectors, in particular the solution sets of SLEs

Closed Sets

A set $S$ is closed under some binary operation if the result is in always in $S$ whenever the arguments are in $S$

• e.g. the set $\mathbb{N}$ of all natural numbers is closed under addition as the sum of any natural number will always be a natural number
• But the set $\mathbb{N}$ is not closed under subtraction as the subtraction between two natural numbers may not be a natural number
  • e.g. a negative number

Asymmetry

• Proof and disproof are not symmetric:
  • Proof (must work for all choices)
  • Counterexample (just need one example)

Subspace

• A set of vectors $S\subseteq {\mathbb{R}}_n$ is a subspace if:

  • $(0,0,0,...,0)\in S$
    • Easy geometric check → pass through origin
  • $S$ is closed under vector addition
  • $S$ is closed under scalar multiplication

• Subspace vs Subset:

  • Subset (no structure)
  • Subspace (lots of structure)

You will need to use proofs/counterexamples to prove/disprove whether a vector subset is a subspace. Worked examples are on the annotated slides. Work through each subspace check from above.

Proving Subspaces

• Focus on proving that a set is closed under addition/multiplication
  • Define variables $u$ and $v$ such that $u,v\in S$
    • $S$ can be any set
    • e.g. $S$ is the solutions to $2x-3y+z=0$
  • State $S(u)=c$ and $S(v)=c$ $b/c$ $u,v\in S$
    • e.g. $S(u)=0$
  • Test $S(u+v)$ and use your earlier statements to prove it equals $c$
    • If you can prove it equals $c$, the set is closed!

Subspaces of ${\mathbb{R}}^2$

• Straight lines that pass through zero are subspaces
  • Gradient of a line is constant, thus addition/multiplication of any point will always result on another point on the line!
• ${\mathbb{R}}^2$ is a subspace
• $S=\{(0,0)\}$ is a subspace

A linear combination of two vectors $u$ and $v$ is: $\alpha u+\beta v$where $\alpha ,\beta \in \mathbb{R}$

Span

• The span is the set of all linear combinations that can be formed from a set of vectors
  • The span of anything is a subspace
• Given $T=\{v_1,v_2,...v_k\}$, the span of $T$ is $span(T)=\{{\alpha }_1{v}_1+{\alpha }_2{v}_2+...+{\alpha }_k{v}_k:a_i\in \mathbb{R}\}$ The span of $T$ is the smallest subspace containing $T$
• $span(\varnothing )=\{0\}$

Testing if something lies within a span is just using a system of linear equations (see annotated slides)

Two Types of Questions

• Find the span of $T$
  • You are given $T$ and have to find $S$
• Find a spanning set of $S$
  • You are given $S$ and have to find $T$

Proving a Spanning Space

• Given subspace $S$, how do we prove $T$ is its spanning space?
  • Must show that every linear combination of the vectors are in $S$
  • Must show every part of $S$ can be created using vectors in $T$
• Watch lecture recording

Find a Spanning Set

To find a spanning space of a subspace, choose vectors from within the subspace until you can recreate the set with only that set of vectors. You will need a vector that can influence each dimension. Thus, for a set with an $x,y$ and $z$ component, your spanning set must have a vector with an $x,y$ and $z$ component!

Because a subspace is closed under addition and multiplication, you can choose any vector without the worry of your spanning space going beyond the subspace. When choosing a vector, don’t choose one that you can already make with the vectors you have already chosen. Try and use the least amount of vectors possible!

Linear Dependence

• A set of vectors is linearly dependent if one of them is a linear combination of the others
• A set of vectors is linearly independent if it is not linearly dependent

Linear Independence Test

In ${\mathbb{R}}^n$, as set $T=\{v_1,v_2,...v_k\}$ is linearly independent if and only if the homogeneous system of $n$ linear equations: ${\alpha }_1{v}_1+{\alpha }_2{v}_2+\cdot \cdot \cdot +{\alpha }_k{v}_k=0$
in the unknowns ${\alpha }_1,{\alpha }_2,...,{\alpha }_k$ has the unique solution: ${\alpha }_1={\alpha }_2=...={\alpha }_k=0$

• This can be tested through using Gaussian Elimination

Properties of Dependence

• A subset of a linearly independent set is linearly independent
• A superset of a linearly dependent set is dependent

Having a dependent set means there is a redundancy

• If a set of vectors $S$ is a dependent set and $v_1$ is a linear combination of the other vectors, then $span(S)=span(S\text{textbackslash}\{v_1\})$
  • You can remove that vector with no change to the span

Bases

• A basis of a subspace contains just enough vectors to be the spanning set of the subspace
  • A basis must be linearly independent
• A basis is the easiest way to specify which vectors are in a subspace