Practice question · Select all that apply
Select every set that IS a vector space under the usual operations.
Hints
- Conditions of the form 'something equals 0' survive addition and scaling.
- Check whether the zero object satisfies each condition, then check scaling by -1.
Show the answer
- B. All infinite sequences of real numbers
- C. All 2x2 matrices whose entries sum to 0
- E. All polynomials p with p(1) = 0
Why
Entry-sum 0 and p(1) = 0 are conditions that equal zero, so they are preserved by sums and scalings; sequences add and scale termwise. Entry-sum 1 excludes the zero matrix, and 'positive leading coefficient' is destroyed by multiplying by -1, and also excludes the zero polynomial.
Practise Vector Spaces
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