Practice question · Select all that apply
T is a linear map from five-dimensional space into three-dimensional space. Select every statement that MUST be true.
Hints
- The image sits inside the codomain, which bounds the rank.
- Feed that bound into Rank-Nullity to bound the nullity from below.
Show the answer
- B. The rank of T is at most 3
- C. T can be surjective
- D. Rank plus nullity equals 5
- E. The nullity of T is at least 2
Why
The image lives in a 3-dimensional codomain, so rank is at most 3 and nullity is therefore at least 5 - 3 = 2. A nonzero kernel forbids injectivity, but a map of rank 3 does exist, so surjectivity is possible. Squeezing a 5-dimensional space into 3 dimensions must collapse something.
Practise Kernel, Image, and Rank–Nullity
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