Practice question · Sort into groups
A vector space must be closed under addition and scaling and must contain a zero vector. Sort each set, taken with its usual operations.
Groups: Is a vector space · Is not a vector space
- All polynomials of degree exactly 3
- All 2x2 matrices
- All pairs (x, y) with x + y = 0
- All functions from the reals to the reals
- All polynomials of degree at most 3
- All pairs (x, y) with x + y = 1
Hints
- Test the cheapest axiom first: does the set contain a zero object?
- For 'degree exactly 3', add to and look at the degree of the result.
Show the answer
Is a vector space: All polynomials of degree at most 3, All 2x2 matrices, All pairs (x, y) with x + y = 0, All functions from the reals to the reals
Is not a vector space: All polynomials of degree exactly 3, All pairs (x, y) with x + y = 1
Why
Degree-exactly-3 polynomials fail twice: the zero polynomial is missing, and plus has degree 1. The set x + y = 1 misses (0, 0). Everything else is closed under addition and scaling, so all the remaining axioms are inherited.
Practise Vector Spaces
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