Proof of the Properties of vector arithmetic
Proof of a(v^{→} + w^{→}) = av^{→} + aw^{→}
We will begin with the two vectors, v^{→} = (v_{1} , v_{2} ,..., v_{n})and w? = w_{1} , w_{2} ,..., w_{n}) and yes we did mean for these to every have n components. The theorem works for general vectors thus we may also do the proof for general vectors.
Here now, as illustrated above this is pretty much just a "computational" proof. What that means is that we'll calculate the left side and then do some basic arithmetic on the result to illustrate that we can make the left side act like the right side. Here is the work.
a(v^{→} + w^{→}) = a {(v_{1}, v_{2}, ... v_{n}) + (w_{1},w_{2},...,w_{n})}
= a (v_{1} + w_{1}, v_{2} + w_{2}, ... v_{n} + w_{n})
= {a (v_{1}+w_{1}), a (v_{2}+w_{2}), .... a (v_{n} + w_{n})}
= {av_{1} + aw_{1}, av_{2} + aw_{2}, ..., av_{n} + aw_{n}}
= {av_{1}, av_{2}, ... , av_{n}}+ {aw_{1}, aw_{2}, ..., aw_{n}}
= a ( v_{1}, v_{2}, ... v_{n}) + a (w_{1},w_{2},...,w_{n})}
= av^{→} + aw^{→}