Wasn't really able to find something here or on Google which answers my question. I am asked to prove the distributive property of vectors such that $$(r + s) * \vec{a} = r * \vec{a} + s * \vec{a}$$ (r and s not being vectors but scalars). The textbook explains the proof of another form of the distributive property, namely: $$r*(\vec{a} + \vec{b}) = r * \vec{a} + r * \vec{b}$$Using the fact that you can add vectors, multiply a vector by a scalar and distribute constants, they showed how to prove this property and it made sense.
The property I have mentioned above that I am not able to prove is left as an exercise to the reader but I am not sure how to solve it. Adding vectors won't help me much since there is only one vector, and I'm not sure what to do with the two constants in the parentheses. Thanks in advance for any help! So far I have: $$(r+s)\begin{pmatrix}a_1 \\ a_2\end{pmatrix} = ?$$ (trying this out first with a 2D vector)
You're almost there, you just have to use the definition of multiplication by scalar. First you get $$(r+s)\begin{pmatrix}a_1 \\ a_2\end{pmatrix} = \begin{pmatrix}(r+s)a_1 \\ (r+s)a_2\end{pmatrix} = \begin{pmatrix}ra_1 + sa_1 \\ ra_2+sa_2\end{pmatrix}$$
where the last equality follows from the usual distributive property of real number arithmetic. Can you finish from here?