Proving Consistency of Ax=b: A Fundamental Property of Matrices

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Homework Statement


Let A be mxn prove Ax=b is consistent for all b if and only if rank(A) = m

Homework Equations


See above

The Attempt at a Solution



I do not know where to begin on this. Any hints would be great
 
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What does it mean for a system to be consistent?

If you augment matrix A and b to get [A|b] and you row reduce, how many rows of non-zeros would you need to get a consistent solution?
 

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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