MATH1012 - Lecture 3

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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

Week 2 material started early due to loss of next Monday’s lesson

Skeleton Slides

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

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.

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MATH1012 - Lecture 4