Linear System Solutions and the Role of Scalar Multiplication

golriz
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I have a question:
If the vectors of v & u are solutions of the nonhomogeneous linear system Ax=b, then ru+sv is a solution of the nonhomogeneous system for any real values of r and s.
is this statement true?
is this statement true for homogeneous systems too?
 
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Hint: When a nonhomogeneous linear system has infinite solutions, then the general solutions can be written in parametric vector form.

This can be used to answer both questions.
 
You should be able to find a counterexample easily for the first question. For the second, try writing the matrix equation for Ax=b using ru+sv.
 

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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