Created on 2024-06-20Asked by Layla Hill (Solvelet student)
Determine if the set W={(x,y,z)∈R3∣x+y+z=0} is a subspace of R3.
Solution
To determine if the set W={(x,y,z)∈R3∣x+y+z=0} is a subspace of R3: 1. **Check if W contains the zero vector:** (0,0,0)∈W because 0+0+0=0. 2. **Check if W is closed under addition:** Let u=(x1,y1,z1) and v=(x2,y2,z2) be in W. u+v=(x1+x2,y1+y2,z1+z2).(x1+y1+z1)+(x2+y2+z2)=0+0=0⟹x1+x2+y1+y2+z1+z2=0⟹u+v∈W. 3. **Check if W is closed under scalar multiplication:** Let c∈R and u=(x,y,z)∈W. cu=c(x,y,z)=(cx,cy,cz).c(x+y+z)=c⋅0=0⟹cx+cy+cz=0⟹cu∈W. 4. **Result:** Since W contains the zero vector, and is closed under addition and scalar multiplication, W is a subspace of R3. Solved on Solvelet with Basic AI Model
Some of the related questions asked by Harper Taylor on Solvelet
DefinitionIn linear algebra, a subspace is a vector space that is inside another vector space. For instance, take the subspace of R2 consisting of all vectors of the form (x,2x).
