Tensor product of kets and bras My question is about tensor products. I have learned that the tensor product between two operators is, for example, $A{\otimes}B$. Suppose we have a system $A$ and a system B. Why we can write the bra of single system A as $\langle{a}\rvert_A$=$\langle{a}\rvert_A$$\otimes$$\mathbb{1} $?
Isn't this representation just for operators?
 A: We have a pair of state spaces $A$ and $B$, their combined state space is the tensor product $A\otimes B$. Its elements are of the form $|a\rangle\otimes|b\rangle$ and (limits of) sums of such elements. 
Bra's are elements of the dual space $A^\ast$, which are (continuous) linear maps $A\to\mathbb C$. Lets write $\textrm{End}(B)$ for the space of all endomorphisms of $B$ (i.e. operators from $B$ to itself). Then $\langle a|\otimes\mathbf1\in A^\ast\otimes\textrm{End}(B)$.
Now let's see what it does on an element of $|\psi\rangle\otimes |b\rangle\in A\otimes B$:
$$\langle a|\otimes\mathbf{1}(|\psi\rangle\otimes|b\rangle) = \langle a|\psi\rangle\otimes|b\rangle.$$
In other words, $\langle a|\otimes\mathbf{1}$ acts exactly as $\langle a|$ on the factor $A$ and does nothing on $B$, i.e. as if $B$ were not really there. In that sense this element can be identified with the corresponding bra of the single system $A$.
A: The representation is for vectors too.
Think of it this way: the formalism is for matrices, of any size. Vectors are matrices with a single column. 
