Mr. Integral

THE BIG IDEA

Vector Spaces

How many independent directions does a model really have?

A long list of vectors can contain redundant information. Span, independence and basis distinguish what the vectors can produce from how many directions are actually needed to describe it.

Begin the first lesson →See the lesson route ↓

3 lessons · No deadlines or locked lessons

SEE THE CONNECTIONS

How the ideas fit together.

A map of the main ideas. Follow a node to its lesson.

  1. Start with01Span & basis
  2. closed families of combinations form02Subspaces
  3. counting independent directions gives03Rank & nullity

YOUR LEARNING ROUTE

One idea at a time.

Read the explanation, use the visualization, then try the check before opening its reasoning.

Keep the unit cheat sheet handy →
Lessons in this unit3 lessons
  1. 01Span, Independence and BasesIdentify redundant vectors and select coordinates relative to an independent spanning set.
  2. 02Subspaces, Null Spaces and Column SpacesCheck closure and distinguish homogeneous solution spaces from translated solution sets.
  3. 03Rank, Nullity and the Four Fundamental SpacesCount independent constraints and locate each fundamental space in its correct ambient space.