Answer to Question #115331 in Linear Algebra for Sourav mondal

Question

Let W {(x, y, z) R3: x + y + z = 0}. Check
if W is a subspace of R3 . Find a non-zero
subspace U of R3 so that W intersection U = (0).

Answer 1

a) Is W a subspace of R3?

if x + y + z = 0, that W {(-y-z, y, z) }

let's check two conditions

1) ∀ x,y∈ W, (x+y)∈ W

2) ∀ x∈W ,α∈R α*x∈W 

1)∀ a₁,a₂∈W

a₁={x₁, y₁, z₁}={-y₁-z₁, y₁, z₁ }

a₂={x₂, y₂, z₂}={-y₂-z₂, y₂, z₂ }

a₁+a₂={-y₁-z₁+(-y₂-z₂), y₁+y₂, z₁+z₂ }

x + y + z = 0:

-y₁-z₁+(-y₂-z₂)+ y₁+y₂+ z₁+z₂=0 the condition is satisfied

2)∀ a₁∈W α*x∈W 

αa₁={αx₁, αy₁, αz₁}={α(-y₁-z₁), αy₁, αz₁ }

α(-y₁-z₁)+αy₁+αz₁= α((-y₁-z₁)+y₁+z₁)=0 the condition is satisfied

b) U-?

U {(x, y, z) R3: f}

where f the equation of the straight line which is passing through (0,0,0) and not lying in the x+y+z=0 plane

for example U {(x, y, z) R3: x=y=z}