Practice question · Multiple choice
A linear map from ℝ⁵ to ℝ³ can never be injective. Which theorem settles that immediately, and how?
Hints
- The image sits inside ℝ³, so how large can the rank be?
- Rank + nullity = 5. Fill in the largest possible rank.
Show the answer
A. Rank-Nullity: rank is at most 3, so the kernel is non-trivial
Why
The image lives in ℝ³ so rank ≤ 3, and rank + nullity = 5 forces nullity ≥ 2, a kernel that big means whole directions collapse to zero. Squeezing five dimensions into three must lose something, and Rank-Nullity is that intuition made exact.
Practise Kernel, Image, and Rank–Nullity
The app has 7 more questions on this lesson, and keeps your place in the course. Mathematics I is free to start.