How do you prove that nullity plus rank equals the dimension of the domain for a finite-dimensional linear map?
Conditions
- The linear map is defined on a finite-dimensional domain.
- A basis for the kernel has been identified.
Reasoning, step by step
- Start with a basis for the kernel.
- Extend this basis to form a complete basis for the domain.
- Identify the added basis vectors that are not in the kernel.
- Apply the linear map to these added vectors.
- Verify that their images form a basis for the range.
- Sum the number of kernel basis vectors (nullity) and range basis vectors (rank).
- Conclude that this sum equals the total dimension of the domain.
Example
Extend a basis of the kernel to a basis of the finite-dimensional domain. Images of the added basis vectors form a basis of the range, giving nullity+rank=domain dimension.
Common misconceptions
- Assuming the images of all domain basis vectors are linearly independent.
- Forgetting that kernel vectors map to zero and do not contribute to the range basis.
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