I'm new to linear algebra and curious about the two properties of a linear transformation:
1) $f(x_i) + f(x_j) = f(x_i+ x_j)$
2) $cf(x) = f(cx)$ where $c$ is some scalar constant
Do either of these two properties entail the other one? I was thinking about it abstractly, and it seems like the linear transformation means that any step you take in the domain is identical in the co-domain.
Or, I can stretch and shift in the domain and that movement will appear exactly the same in the co-domain.
Or, the mapping preserves addition (subtraction) and multiplication (division).
I had two ways to try and answer this question, but got stuck.
First I asked myself if the addition of any two arbitrary reals a, b could be reached by multiplication of any two other arbitrary reals, c, d. In some vague sense I thought this would help me see if there was a link between (1) and (2).
It occurred to me that you can set c = (a+b) and d = 1 to achieve c * d = a+b for all reals a, b. It occurred to me that even if you prohibit using the identity you can simply set c = 1/(a+b) and d = (a+b)2 and this works so long as we avoid (a+b)=0. To me this felt like an additive "shift" (a+b) could yield the same location as a "stretch" (c*d). However, this did not feel rigorous.
For my second attempt I tried to find a counter-example, where one of the two linear transformation properties was preserved but the other was not, but came up empty handed.
Help appreciated, thank you!
EDIT: I assumed domains and co-domains in the reals. Not sure how this changes things.
EDIT 2: It has been claimed that these conditions are equivalent only over the reals. Does anyone have a proof of their equivalence over the reals?
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