Confusing XOR notation in textbook sample solution
The notation $c=\oplus~c_i$ is (terrible) shorthand for $$c=\bigoplus_{i \in I(c)} c_i$$ where the sum sign should be replaced by the big xor sign which could also be written as $$ c=\sum_{i \in I(c)} c_i,$$where $\sum$ denotes vector addition modulo 2. 

An example of this decomposition (for length 8 vectors) is
$$c=(1,0,1,0,0,0,1,0)=$$
which is nonzero in positions 1,3, and 7, and satisfies
$$c=(1,0,0,0,0,0,0,0)\oplus(0,0,1,0,0,0,0,0)\oplus(0,0,0,0,0,0,1,0)$$
$$=c_1\oplus c_3 \oplus c_7=\bigoplus_{i \in I(c)} c_i$$ where $I(c)=\{1,3,7\}.$ 

Then, since $E$ is assumed to be linear, all the places you marked with an arrow make sense.