# X circle plus Y = R^3

• Sep 10th 2011, 09:31 AM
dwsmith
X circle plus Y = R^3
Let $\displaystyle a,b,x\in\mathbb{R}$ not all equal to zero and $\displaystyle X=\{(x,y,z)\in\mathbb{R}^3: ax+by+cz=0\}$

Find a subspace Y such that $\displaystyle \mathbb{R}^3= X\oplus Y$ and deduce X is 2 dimensional.

I am lost on this one.

Note:

$\displaystyle \oplus$ is the external direct sum define as $\displaystyle X\oplus Y=(x_1,y_1)+(x_2,y_2)=(x_1+x_2,y_1+y_2)$
$\displaystyle \lambda (x_1,y_1)=(\lambda x_1,\lambda x_2)$

If $\displaystyle X \cap Y =\{0\}$, then $\displaystyle \oplus$ can be define as $\displaystyle \text{dim}(X+Y)=\text{dim}X+\text{dim}Y$ that is the internal direct sum $\displaystyle X\oplus Y$
• Sep 10th 2011, 10:07 AM
HallsofIvy
Re: X circle plus Y = R^3
It should be obvious that this is a plane and that the vector $\displaystyle a\vec{i}+ b\vec{j}+ c\vec{k}$ is perpendicular to the plane.
• Sep 10th 2011, 12:01 PM
dwsmith
Re: X circle plus Y = R^3
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Quote:

Originally Posted by HallsofIvy
It should be obvious that this is a plane and that the vector $\displaystyle a\vec{i}+ b\vec{j}+ c\vec{k}$ is perpendicular to the plane.

Ok so I just need to add the subspace of the normal vector to the plane. How can I describe the set Y to encompass all the normal vectors <a,b,c> to the plane?

$\displaystyle Y=\{(x,y,z)\in\mathbb{R}^3: \text{???}\}$
• Sep 10th 2011, 02:41 PM
HallsofIvy
Re: X circle plus Y = R^3
Y is the subspace spanned by $\displaystyle a\vec{i}+ b\vec{j}+ c\vec{k}$. That is the space of all vectors of the form $\displaystyle \alpha(a\vec{i}+ b\vec{j}+ c\vec{k})$ where $\displaystyle \alpha$ can be any real number.