Rank-Nullity Theorem
for a linear map with finite-dimensional.
Theorem 3.4 (Rank-Nullity Theorem)
Let be a linear map, where is finite-dimensional. Then,
Proof
Let and ; since , we have .
If , then , so is the zero map, and . Therefore .
If , let be a basis of , so that for all . Extend it to a basis of ; it suffices to show that is a basis of .
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Spanning. For , pick with and write . By linearity,
so lies in the span of the images.
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Linear independence. Suppose . By linearity, for , so . Writing and comparing with the unique representation of in the basis of gives .
Examples
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Zero linear map. and , so .
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Identity map. and , so .
Related
Stated in
- Theorem 3.4 (Rank-Nullity Theorem)§3.1 Linear Maps