TR3.5

Flo 10

Exr0n 2021-09-27 Mon 12:00

#flo

1 Span

1.1 Smallest/largest containing subspaces

  • Spans are not the largest vector space that contains the given vectors Pasted image 20200924131215.png
  • The span of that vector is a line. It's a subspace. But it's not the biggest, because there's also R2

1.2 Spans tend to be infinite

  • Usually a span has infinitely many vectors (unless you're in a weird field (modulo) or have the zero span)
  • In the span of just one vector, you can multiply by any scalar which there tends to be infinite of Pasted image 20200924131215.png
  • The span of that vector is a line. It's a subspace. But it's not the biggest, because there's also R2
  • It only won't be infinite if your span is the span of \(()\) (empty list)

1.3 Given a linearly independent set of vectors, would the span equal to

the vector space?

:CUSTOMID: given-a-linearly-independent-set-of-vectors-would-the-span-equal-to-the-vector-space

  • No? It's unclear which vector space is being referred to.

1.4 Span of vectors (example 2.6)

  • When it's two vectors, you'd expect the span to be a 2d plane unless the vectors are parallel
    • In other words, if they are linear combinations or scalar multiples of one another
    • A linear combination on one other vector is the same as a scalar multiple
    • in 2space they have to not be colinear, in 3space they have to not be coplanar.
    • They have to be linearly independent
  • That probably generalizes to higher and lower dimensions

1.5 Adding a vector doesn't make the span smaller

  • Because you can just do what you had originally and make it's coefficient zero

1.6 Size of spans/subspaces

  • You can't really just count the number of vectors, because say a line and a plane both have infinite points
  • But we still want a plane to be larger than a line and a space to be larger than a plane
  • So one way we compare is to say \(A\) is larger than \(B\) if \(B\) is strictly contained within \(A\)
  • something like "dimensionality", maybe the minimum number of vectors needed for their span to be equal to the space

2 2.7 Span is the smallest containing subspace

  • First the proof shows that the span is a subspace
  • Then, because the span only neds to contain each vector and be a subspace, any subspace containing those vectors will at least contain the span.

3 Linear Dependence

  • When one of the vectors provides no "new information" aka can be constructed by a linear combination of vectors you already had
  • It's a property of a set of vectors, not just one vector. A single vector is always linearly independent on its own, because there's nothing else to depend on.
  • The span of the zero vector \((0)\) is linearly dependent on itself, and you already don't really get anything. So we usually talk about it as a span of no vectors \(()\)

4 Rotation matrices