Questions: Vector Spaces

Question 1

Which of the following sets is NOT a vector space over ℝ under the standard operations?

• A. All polynomials of degree ≤ 3
• B. All polynomials of degree exactly 3
• C. All continuous functions on [0, 1]
• D. All 2×2 real matrices

Answer: (B)

Polynomials of degree *exactly* 3 fail closure under addition: (x³ + 1) + (−x³ + x) = x + 1, which has degree 1, not 3. A vector space must be closed — adding two elements must produce another element in the set. The other three options satisfy all eight axioms and are genuine vector spaces.

Question 2

The set of all solutions to a homogeneous linear system Ax = 0 always forms a vector space.

• T. True
• F. False

Answer: True

The solution set of Ax = 0 (the null space of A) is always a subspace of ℝⁿ, hence a vector space. If Ax = 0 and Ay = 0, then A(x + y) = 0 and A(cx) = 0, so the solution set is closed under addition and scalar multiplication. The zero vector is always a solution (A·0 = 0), providing the required additive identity.

Question 3

Why do mathematicians define an abstract notion of 'vector space' rather than just working with ℝⁿ directly?

Model answer: The abstract definition captures the essential algebraic structure that ℝⁿ, polynomials, matrices, and function spaces all share. Any theorem proved for abstract vector spaces automatically applies to all of these examples at once, without re-proving it for each case.

This is the power of abstraction. The eight axioms define exactly the structure needed for concepts like linear independence, span, and basis to make sense. Proving theorems at the abstract level gives results that apply simultaneously to geometric vectors, polynomial spaces, and infinite-dimensional function spaces — far broader than ℝⁿ alone.