Exercise · ZW-VECTOR-SPACE-009 · ★★★ Advanced
On \(\mathbb{R}_{>0}\) define \(u\oplus v=uv\) and \(c\odot u=u^c\). What is the zero vector?
Choose one option.
Show answer
- C. \(1\)
Explanation
The zero must satisfy \(u\oplus 0=u\), that is \(u\cdot 0_V=u\), so the zero vector is the number \(1\). The number \(0\) is not even in the set.
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-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?