Mathematics · University
Rank-nullity for a finite-dimensional map
The dimension is that of the domain, not the codomain. Rank cannot exceed the domain dimension.
Method and conditions
The dimension is that of the domain, not the codomain. Rank cannot exceed the domain dimension.
dim ker T = dim domain T − rank T
Worked examples
A linear map T has domain R^15, codomain R^18, and rank 7. What is dim ker T? Round the final result to 4 decimal places if needed.
Rank-nullity in the domain gives 15 = 7 + dim ker T.
8
A linear map T has domain R^4, codomain R^7, and rank 0. What is dim ker T? Round the final result to 4 decimal places if needed.
Rank-nullity in the domain gives 4 = 0 + dim ker T.
4
Try it
A linear map T has domain R^38, codomain R^41, and rank 10. What is dim ker T? Round the final result to 4 decimal places if needed.
A linear map T has domain R^24, codomain R^27, and rank 23. What is dim ker T? Round the final result to 4 decimal places if needed.
Check your work
Answers and marking notes
- 28
Rank-nullity in the domain gives 38 = 10 + dim ker T.
- 1
Rank-nullity in the domain gives 24 = 23 + dim ker T.
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