Linear Algebra › Section B · Matrix algebra › Topic B.8
Subspaces, column space, and null space
Lines and planes through the origin behave like smaller copies of space itself: you can add vectors in them and scale them without ever leaving. Every matrix carries two such sets, one of reachable outputs and one of inputs it sends to zero, and a few well-chosen vectors describe each completely.
In this topic
- Sets that are closed
- Definition of a subspace
- Worked example: testing three sets
- The column space
- The null space
- Two different kinds of description
- Bases
- Finding bases for Nul A and Col A
- Worked example: bases for both spaces
- Common mistakes to avoid
- Key takeaways
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