Linear transformation conditions?

Question

$L(u+v) = L(u) + L(v)$ for every $u$ and $v$
$L(cu) = cL(u)$ for any $u\in V$, and $ c$ any real number

Do both conditions have to meet or can we say a vector is a linear transformation of the other if just one of these conditions meet? Or is it that if one meets, the other also will (which is not likely)

A linear transformation must meet both of those conditions simultaneously. Also, $c$ merely needs to be taken from the associated scalar field which depending on your specific example might or might not be the field of the real numbers. There are plenty of examples where the scalar field is something different than the reals. — JMoravitz, Dec 09 '17 at 00:51
You need both. There are examples of functions that satisfy each but not the other. Perhaps you can look for them. — Ethan Bolker, Dec 09 '17 at 00:51
So as to prevent people from trying and failing to come up with examples where one is satisfied but the other is not, here is a link to a related question and here is a link to another. Such examples will often be rather exotic in nature. — JMoravitz, Dec 09 '17 at 01:08
@JMoravitz Thanks a lot, I was trying to think of an example where one is true and the other isn't, your links provided great examples. — Stannis Baratheon, Dec 09 '17 at 01:15

score 1 · Accepted Answer · answered Dec 09 '17 at 00:55

1

It must meet both conditions for a transformation to be called linear. Meeting one of these conditions doesn't necessarily imply that the other condition will hold.

answered Dec 09 '17 at 00:55

BR Pahari

2,694

Can you provide some simple counter examples or links or problems related to your claim? – Edumaths555 Dec 15 '17 at 16:48
See https://math.stackexchange.com/questions/2211907/map-not-preserving-vector-addition-but-preserving-scalar-multiplication/2211933 – BR Pahari Dec 17 '17 at 14:56

Linear transformation conditions?

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