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

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- Feb 21st 2011, 01:53 PMrabih2011Subspace of R3
Determine weather w={(x,2x,3x): x a real number} is a subspace of R3

- Feb 21st 2011, 02:08 PMDarkprince
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 PMPlato
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 $\displaystyle \mathbb{R}^3$.