# Subspace of R3

• Feb 21st 2011, 01:53 PM
rabih2011
Subspace of R3
Determine weather w={(x,2x,3x): x a real number} is a subspace of R3
• Feb 21st 2011, 02:08 PM
Darkprince
To be a subspace it must confirm three axioms:

Containing the zero vector, closure under addition and closure under scalar multiplication

a) (0,0,0) is contained since (0,2*0,3*0)=(0,0,0)
b) Let (y,2y,3y) be a vector, then (x,2x,3x)+(y+2y+3y)=(x+y,2(x+y),3(x+y)) => closure under addition
c) Let c be a constant, then c*(x,2x,3x)=(cx, 2(cx), 3(cx)) => closure under scalar multiplication

So it's a subspace. I hope this is what you wanted and that will help you!
• Feb 21st 2011, 02:12 PM
Plato
Quote:

Originally Posted by rabih2011
Determine weather W={(x,2x,3x): x a real number} is a subspace of R3

Notice that $\displaystyle (0,0,0)\in \mathbb{R}^3$. WHY?
Suppose that $\displaystyle \{a,b\}\subset W$ and $\displaystyle \alpha\in\mathbb{R}$.
If you can show that $\displaystyle \alpha a+b\in W$ then $\displaystyle W$ is a subspace of $\displaystyle \mathbb{R}^3$.