MATH1012 - Lecture 4

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MATH1012 - Lecture 3
Skeleton Slides
⭐ $b/c$ → because

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 $T$

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MATH1012 - Lecture 5
Annotated Slides