It seems that one is given a real vector space $(V,+,\cdot)$, a set $U$ and a function $f:V\to U$ such that $f(V)=U$, and that the question is to show that $U$ is a vector space when endowed with the addition $\oplus$ and the scalar multiplication $\odot$ defined by
f(u)\oplus f(v)=f(u+v),\qquad r\odot f(u)=f(r\cdot u),
for every $r$ in $\mathbb R$ and $u$ and $v$ in $V$.
In the question, $f$ is denoted $[\ \ ]_w$ (and the meaning of this notation is not given, despite several proddings to this effect in the comments) but this is irrelevant.
On the other hand, a crucial hypothesis is missing, to make sure that these definitions even make sense, for example it is necessary that for every $u$, $u'$, $v$ and $v'$ in $V$ such that $f(u)=f(u')$ and $f(v)=f(v')$, one has $f(u+v)=f(u'+v')$, otherwise the sum $x\oplus y$ of two elements $x$ and $y$ of $U$ might depend on the choice of their preimages $u$ and $v$ in $V$, in which case $\oplus$ would be undefined. Similarly for the scalar multiplication $\odot$.
A way to ensure the soundness of the definition of $\oplus$ and $\odot$ is to assume that $f$ is bijective but this is not the only one. Here again, to know what is $[\ \ ]_w$ would be helpful.
All this being taken care of... it is highly probable that the lecture notes include a general result showing why $(U,\oplus,\odot)$ is a vector space. Otherwise, a direct verification of the axioms is possible, and not difficult.