Question about the conditions for a linear map/transformation

Asked Jul 01 '17 at 12:39

Active Jul 01 '17 at 12:39

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I have been taught and always read that there are two conditions required for a map or transformation to be linear is

$T(\alpha \textbf{u})=\alpha T(\textbf{u})$
$T(\textbf{u}+\textbf{v})=T(\textbf{u})+T(\textbf{v})$

Now my question is, does the second not imply the first? Why are both of these conditions stated? Is there any examples of non-linear mappings in which the second condition is satisfied but the first is not?

asked Jul 01 '17 at 12:39

Meep

3,167

1

The second only implies the first when $\alpha$ is rational. See Cauchy's Functional Equation. – lulu Jul 01 '17 at 12:41
How would the second imply the first exactly? And yes there are examples: https://math.stackexchange.com/questions/2132215/a-real-function-which-is-additive-but-not-homogenous – Nap D. Lover Jul 01 '17 at 12:48
@lulu Thank you. I completely ignored the fact that, even in the above, 2 Only deals with integers. The link is very interesting that it extends to all rationals. – Meep Jul 01 '17 at 13:07
The second condition does not imply the first. Consider $\alpha\in \mathbb{C}$. – mark haokip Jul 04 '17 at 16:12

Question about the conditions for a linear map/transformation

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