Answer to Question #151595 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

Let u = (x₁,y₁,z₁) and v = (x₂,y₂,z₂) be two vectors from W. Then:

(1) u+v = (x₁+x₂, y₁+y₂, z₁+z₂) belongs to W, as

(x₁+x₂) + (y₁+y₂) + (z₁+z₂) = (x₁ + y₁ + z₁) + (x₂ + y₂ + z₂) = 0 + 0 = 0

(2) for any constant a the vector au = (ax₁,ay₁,az₁) belongs to W, as

(ax₁+ ay₁+az₁) = a(x₁ + y₁ + z₁) = a*0 = 0

With (1) and (2) we have proved that W is a vector subspace of R³.

Let V = {(x, y, z) R³ : x = y = 0}. Lets prove that V is a subspace of R³.

Let u = (x₁,y₁,z₁) and v = (x₂,y₂,z₂) be two vectors from V. Then:

(1) u+v = (x₁+x₂, y₁+y₂, z₁+z₂) belongs to V, as

x₁+ x₂ = 0 + 0 = 0 and y₁+ y₂ = 0 + 0 = 0

(2) for any constant a the vector au = (ax₁,ay₁,az₁) belongs to V, as

ax₁= a*0 = 0 and ay₁= a*0 = 0.

With (1) and (2) we have proved that V is a vector subspace of R³.

$W \cap V = {0}$ , because of all vectors from this set satisfies the system of equations:

x + y + z = 0

x = 0

y = 0

It is clearly, that only (0, 0 , 0) satisfies this system.