1. ## Subspace of R3

Determine weather w={(x,2x,3x): x a real number} is a subspace of R3

2. 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!

3. Originally Posted by rabih2011
Determine weather W={(x,2x,3x): x a real number} is a subspace of R3
Notice that $(0,0,0)\in \mathbb{R}^3$. WHY?
Suppose that $\{a,b\}\subset W$ and $\alpha\in\mathbb{R}$.
If you can show that $\alpha a+b\in W$ then $W$ is a subspace of $\mathbb{R}^3$.