Created on 2024-06-20Asked by Abigail Nelson (Solvelet student)
Determine if the set V={(x,y)∣x,y∈R} with the operations of vector addition and scalar multiplication forms a vector space.
Solution
To determine if the set V={(x,y)∣x,y∈R} forms a vector space: 1. **Check the axioms of a vector space:** - **Closure under addition:** For u=(u1,u2) and v=(v1,v2) in V, u+v=(u1+v1,u2+v2) is also in V. - **Closure under scalar multiplication:** For u=(u1,u2) in V and c∈R, cu=(cu1,cu2) is also in V. 2. **Check other axioms:** - **Associativity of addition:** (u+v)+w=u+(v+w). - **Commutativity of addition:** u+v=v+u. - **Existence of additive identity:** There exists 0=(0,0)∈V such that u+0=u for all u∈V. - **Existence of additive inverse:** For every u=(u1,u2)∈V, there exists −u=(−u1,−u2)∈V such that u+(−u)=0. - **Distributivity of scalar multiplication over vector addition:** c(u+v)=cu+cv. - **Distributivity of scalar multiplication over scalar addition:** (c+d)u=cu+du. - **Associativity of scalar multiplication:** c(du)=(cd)u. - **Existence of multiplicative identity:** 1u=u for all u∈V. 3. **Result:** The set V={(x,y)∣x,y∈R} with the operations of vector addition and scalar multiplication forms a vector space as it satisfies all the vector space axioms. Solved on Solvelet with Basic AI Model
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DefinitionA vector is essentially anything with size and direction. We have sets of vectors that satisfy certain axioms and under the field of the same vector space are closed under addition and scalar multiplication, these are known as Vector Spaces. Example: Rn is an n-dimensional vector space.
