Linear transformation / analysis help

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SUMMARY

The discussion centers on the linear mapping T and its properties, specifically the relationship between the inputs p' and p'' and their transformed outputs T(p') and T(p''). The key conclusion is that to satisfy the condition |T(p') - T(p'')| <= 1/10, the difference |p' - p''| must be chosen as |p' - p''| = 1 / (sqrt(10)*10). This demonstrates that the bound for T is tight and cannot be improved, confirming the correctness of the approach taken by the user, Daniel.

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eckiller
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Hi,

I have that

|T(p)| <= sqrt(10)*|p|

where T is a linear mapping. The question is: How small must |p' - p''| be in order that |T(p') - T(p'')| <= 1/10.

This is what I did:

T linear, so

|T(p') - T(p'')| = |T(p' - p'')|.

Applying the bound:

|T(p' - p'')| <= sqrt(10)*|p' - p''|

So pick |p' - p''| = 1 / (sqrt(10)*10).

Then

|T(p' - p'')| = |T(p') - T(p'')| <= 1/10.

Is that right?
 
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Yes, your approach is correct. By using the given bound for T, you were able to find a value for |p' - p''| that would ensure that |T(p') - T(p'')| is less than or equal to 1/10. This shows that the bound given for T is tight and cannot be improved upon. Great job!
 

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