Very simple linear algebra question:

frasifrasi
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This is a very basic question, but I am not understanding the concept here.

it asks, consider the linear transformation:

y_1(y subscript 1) = x_2 - x_3
y_2 = x_1*x_3
y_3 = x_1 - x_2

Can anyone explain why this is not linear? I am not sure what the criteria is and the book doesn't have any example like this reasoned out.

Thank you.
 
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A transformation T is linear if T(cv) = cT(v) and T(v+w) = T(v) + T(w) for all scalars c and vectors v and w. Does this help?
 
In particular, look at what happens to y2.
 
Ok, so for y_2:

x(x_1*x_3) is not the same as...?

I am not sure how to apply this. Can someone explain how to proceed?
 

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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