Linear Algebra - Find a subspace

cristina89
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Homework Statement



V is a subspace of the vector space R^{3} given by:
V = {(x, y, z) E R^{3} / x + 2y + z = 0 and -x + 3y + 2z = 0}
Find a subspace W of R^{3} such that R^{3} = V\oplusW

I'm really lost in this. My teacher didn't give any example of how to solve this kind of exercise... Can anyone help me how to start and develope this?
 
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each of those equations defines a plane - what is the intersection of two planes?
 
lanedance said:
each of those equations defines a plane - what is the intersection of two planes?

A straight line?

Well, if I solve this system:

x = -2y + z
y = -3/5z --> x = -11/5z
 
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To start with, do you know what your looking for ? Do you know what is the direct sum of two subspaces?
 
So the subspace V is a line through the origin, and can be represented by the span of a single vector parallel to that line.

Now onto Dansure's question...
 
W will be a plane perpendicular to that line, containing the origin.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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