Practice question · Multiple choice
Rank-Nullity says rank + nullity = dim(domain), with no mention of the codomain. Why does the theorem account for the DOMAIN's dimension rather than the space the map lands in?
Hints
- Ask what each dimension of the input can do: get flattened, or come through.
- Map three-dimensional space into a hundred-dimensional one. What does the size of the target change?
Show the answer
C. Because it is bookkeeping on the input side only.
Why
The accounting happens on the input side: every dimension of the domain is either crushed to zero or survives into the image, and the two must sum to what you started with. The codomain is room rather than constraint, map three dimensions into a hundred and the image is still at most three-dimensional. A map from five to three has nullity at least 2, so it cannot be injective.
Practise Kernel, Image, and Rank–Nullity
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