Mathematics · University
Bases, rank and linear maps · revision notes
A basis both spans a vector space and is linearly independent.
Key ideas and conditions
Basis
A basis both spans a vector space and is linearly independent.
Rank
The rank is the dimension of the image, or equivalently the number of pivot columns.
Nullity
Rank plus nullity equals the dimension of the domain.
Subspace test
A candidate subspace must contain zero and be closed under addition and scalar multiplication.
Explain without looking
- A linear map from R^4 to R^4 is injective. What is its rank?
- Prove that the kernel of a linear map is a subspace.
Further reading and review references
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