Why can a linear map from a five-dimensional space to a three-dimensional space not be injective?
Conditions
- The domain is five-dimensional.
- The codomain is three-dimensional.
- The map is linear.
Reasoning, step by step
- Recall that injectivity requires the kernel to be trivial (nullity zero).
- Apply the rank-nullity theorem: nullity + rank = domain dimension.
- Note that the rank cannot exceed the dimension of the codomain (3).
- Calculate the minimum nullity: .
- Conclude that since nullity is at least 2, the kernel is non-trivial.
- Therefore, the map cannot be injective.
Example
A linear map from a five-dimensional space to a three-dimensional space cannot be injective: its rank is at most three, so its nullity is at least two.
Common misconceptions
- Believing that a larger domain always allows for injectivity.
- Confusing surjectivity with injectivity in the context of dimension constraints.
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