A web page was found that was very helpful. We have nothing but respect for the page.
Something on the page deserved a different perspective.
Reading through the page, it seemed (see Appendix A) fairly certain that they were saying to prove linearity you need to prove the following three things (in the equations below, a,b are scalars and u,v are the math object of interest):
- T(av) = aT(v)
- T(u+v) = T(u) + T(v)
- T(av + bu) = T(av) + T(bu) = aT(v) + bT(u)
Our argument is that the third is built on the first two and that if the third is true then we have the first two by implication.
Observe what happens if we choose b=0:
- T(av + bu) = T(av) + T(bu) = aT(v) + bT(u)
- T(av + 0) = T(av) + T(bu) = aT(v) + 0T(u)
- T(av + 0) = T(av) + 0 = aT(v) + 0
- T(av) = T(av) = aT(v)
- We can get to the first one by starting with the third
Observe what happens if we choose a=1 and b=1:
- T(av + bu) = T(av) + T(bu) = aT(v) + bT(u)
- T(1v + 1u) = T(1v) + T(1u) = 1T(v) + 1T(u)
- T(v + u) = T(v) + T(u) = T(v) + T(u)
- We can get to the second one by starting with the third
Now that we know this, we can declare linearity by proving T(av + bu) = aT(v) + bT(u).
Appendix A
If you keep going further into the presentation you reach a point where they only mention the third equation to prove linearity.
Vectoreverything 