Practice question · Select all that apply
Select every subset of the plane that DOES contain the zero vector but is still NOT a subspace.
Hints
- First discard anything that fails the zero-vector test, the question excludes those.
- For the two axes, add (1, 0) to (0, 1). For the cubic, add (1, 1) to (2, 8).
Show the answer
- A. The union of the x-axis and the y-axis
- B. All (x, y) with y equal to x cubed
- D. The closed first quadrant, where neither coordinate is ever negative
Why
The two axes contain 0 and are closed under scaling, yet (1, 0) + (0, 1) = (1, 1) lies on neither axis. The first quadrant contains 0 and is closed under addition, but not under scaling by -1. The cubic contains 0 but (1, 1) + (2, 8) = (3, 9), and 9 is not 27. The line y = 2x is a genuine subspace, and y = 2x + 3 was excluded because it misses the origin.
Practise Subspaces
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