Exercise · ZW-VECTOR-SPACE-001 · ★☆☆ Basic
With the usual addition and scalar multiplication, which of these sets is a vector space over \(\mathbb{R}\)?
Choose one option.
Show answer
- C. Polynomials of degree at most 2
Explanation
Polynomials of degree \(\le 2\) are closed under addition and scalars and contain zero. Those of degree exactly 2 are not (\((t^2+t)+(-t^2+1)=t+1\)), the first quadrant fails with negative scalars, and \(\mathbb{Z}^2\) fails with scalars such as \(1/2\).
Review the concept: Vector space
A real vector space is a set \(V\) with an addition and a scalar multiplication by reals satisfying eight axioms; its elements are called vectors, whether arrows, matrices or functions.
\[u+v\in V,\quad cv\in V,\quad (c+d)v=cv+dv\]
More practice problems: vector space
- ZW-VECTOR-SPACE-002 In the space of 2× 2 matrices, which is the zero vector?
- ZW-VECTOR-SPACE-003 If v=(2,-3,1) in ℝ^3, what is its additive inverse -v?
- ZW-VECTOR-SPACE-004 If v=(4,-7), what is 0· v?
- ZW-VECTOR-SPACE-005 In ℝ^2 we use the usual addition and the product codot(x,y)=(cx, y). Which property fails?
- ZW-VECTOR-SPACE-006 Let S=\f:ℝ→ℝ| f(0)=1\. If f,g∈ S, what is (f+g)(0) and what does it imply?
- ZW-VECTOR-SPACE-007 With the usual operations of ℝ^2, which of these subsets is a vector space?
- ZW-VECTOR-SPACE-008 In any vector space V, what does (-1)· v equal?
- ZW-VECTOR-SPACE-009 On ℝ_>0 define uoplus v=uv and codot u=u^c. What is the zero vector?