= Solution
For binary <linear codes> of length $n$, the <bar product of binary linear codes> is $C_1|C_2=\{(u,u+v):u\in C_1,v\in C_2\}$. The parametrization is <injective> and linear, so its <dimension> is $k_1+k_2$. If $v=0$, a nonzero word has <Hamming weight> at least $2d_1$; if $v\ne0$, $\operatorname{wt}(u)+\operatorname{wt}(u+v)\ge\operatorname{wt}(v)\ge d_2$. Taking $(u,u)$ and $(0,v)$ of minimum weight shows
$$
\boxed{\dim(C_1|C_2)=k_1+k_2,\qquad d(C_1|C_2)=\min(2d_1,d_2).}
$$
For $a\in C_2^\perp,b\in C_1^\perp$, the inner product of $(u,u+v)$ and $(a,a+b)$ is $u\cdot b+v\cdot a+v\cdot b=0$, since $C_2\subseteq C_1$. The spaces have complementary <dimensions>, proving $(C_1|C_2)^\perp=C_2^\perp|C_1^\perp$.
Define the <Reed-Muller code> $\operatorname{RM}(d,r)$ by evaluating all multilinear <polynomials> of total degree at most $r$ at the $2^d$ points of $\mathbb F_2^d$. The squarefree monomials are independent functions, giving dimension $\sum_{j=0}^r\binom dj$. Splitting according to the last variable gives the <Reed-Muller bar-product recursion> $\operatorname{RM}(d,r)=\operatorname{RM}(d-1,r)|\operatorname{RM}(d-1,r-1)$, with the endpoint codes interpreted in the usual way. The distance recursion gives $2^{d-r}$.
To identify the <Dual of a Reed-Muller code>, multiply a monomial of degree at most $r$ by one of degree at most $d-r-1$. The product omits some variable, so its sum over $\mathbb F_2^d$ is even and its binary inner product is zero. The binomial dimension identity then gives
$$
\boxed{\operatorname{RM}(d,r)^\perp=\operatorname{RM}(d,d-r-1)\quad(0\le r<d).}
$$
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