Span of an infinite set. Exam Question.

heshbon
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Homework Statement



Suppose V=<Z> where Z is an infinite set, So Z spans V.
Suppose that every pair of distinct elements of Z is linearly independent.
Is it possible that V is finite dimensional? Justify your answer.

Homework Equations





The Attempt at a Solution


All the vectors must be linearly independent if any two are so...
I don't think you can as a basis for the space would have infinite linaerly independent vectors.
 
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All of the vectors don't have to be linearly independent just because every pair is. Let Z be the set of all unit vectors in the first quadrant in the two dimensional plane. Since they all point in different directions, any pair is linearly independent.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

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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