Let's say I have a vector(adsbygoogle = window.adsbygoogle || []).push({}); xin [tex]\mathcal{R}^3[/tex]. Let's also suppose that any vectorxundergoes the transformationx'=kx(wherekis a positive real).

Obviously, normalizing the vector will give us a quantity which isinvariantto uniform scaling. In fact, [tex]\frac{\mathbf{x'}}{|\mathbf{x'}|} = \frac{k\mathbf{x}}{|k\mathbf{x}|} = \frac{\mathbf{x}}{|\mathbf{x}|}[/tex].

Now, is it possible to find another quantity which is invariant to bothtranslationanduniform-scaling?

The vector x would now undergo the transformationx'=kx+a, where k is a constant scalar, and a is a constant vector in [tex]\mathcal{R}^3[/tex].

Thanks!

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# Scaling and Translation Invariance

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