Exercise · ZW-VECTOR-SPACE-006 · ★★☆ Intermediate
Let \(S=\{f:\mathbb{R}\to\mathbb{R}\mid f(0)=1\}\). If \(f,g\in S\), what is \((f+g)(0)\) and what does it imply?
Choose one option.
Show answer
- B. It equals \(2\); \(S\) is not closed under addition
Explanation
\((f+g)(0)=f(0)+g(0)=1+1=2\ne 1\), so \(f+g\notin S\) and \(S\) is not a vector space.
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-001 With the usual addition and scalar multiplication, which of these sets is a vector space over ℝ?
- 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-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?