Exercise · ZW-VECTOR-SPACE-005 · ★☆☆ Basic
In \(\mathbb{R}^2\) we use the usual addition and the product \(c\odot(x,y)=(cx,\,y)\). Which property fails?
Choose one option.
Show answer
- C. \((c+d)\odot v=c\odot v+d\odot v\)
Explanation
With \(v=(x,y)\): \((c+d)\odot v=((c+d)x,\,y)\) but \(c\odot v+d\odot v=((c+d)x,\,2y)\). They differ when \(y\ne 0\). The other three hold.
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-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?