Linear Algebra › Section D · Vector spaces › Topic D.6
Rank and the rank-nullity theorem
The rows and the columns of a matrix span different spaces, yet those spaces always have the same dimension, the rank. The rank-nullity theorem then balances the directions a matrix keeps against the directions it sends to zero.
In this topic
- Three spaces from one matrix
- The row space and its basis
- Worked example: bases for all three spaces
- Rank: row rank equals column rank
- The rank-nullity theorem
- Worked example: using the theorem
- Full rank
- Rank and the invertible matrix theorem
- Rank in practice
- Common mistakes to avoid
- Key takeaways
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