Created on 2024-06-20Asked by Mateo Anderson (Solvelet student)
Determine if the set W={(x,y)∣x+y=0} is a subspace of V={(x,y)∣x,y∈R}.
Solution
To determine if the set W={(x,y)∣x+y=0} is a subspace of V={(x,y)∣x,y∈R}: 1. **Check if W is non-empty:** 0=(0,0)∈W⟹0+0=0 (satisfied). 2. **Closure under addition:** u=(u1,u2),v=(v1,v2)∈W⟹u1+u2=0,v1+v2=0.u+v=(u1+v1,u2+v2)⟹(u1+v1)+(u2+v2)=(u1+u2)+(v1+v2)=0+0=0.u+v∈W. 3. **Closure under scalar multiplication:** u=(u1,u2)∈W,c∈R⟹u1+u2=0.cu=(cu1,cu2)⟹cu1+cu2=c(u1+u2)=c⋅0=0.cu∈W. 4. **Result:** The set W={(x,y)∣x+y=0} is a subspace of V={(x,y)∣x,y∈R}. Solved on Solvelet with Basic AI Model
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DefinitionA Collection containing all the vectors called Vector spaces under addition and scalar multiplication. In other words, subspaces are just a set of vector which also forms another vector space. Example: The set of all vectors is a vector space The set of all vectors of the form (x, 0, 0) is a subspace
