Tensor Product of Pauli Matrices

[oshilinawa]

Homework Statement

Suppose that $$[\sigma_a]_{ij}$$ and $$[\eta_a]_{xy}$$ are Pauli matrices in two different two dimensional spaces. In the four dimensional tensor product space, define the basis:
$$|1\rangle=|i=1\rangle|x=1\rangle$$
$$|2\rangle=|i=1\rangle|x=2\rangle$$
$$|3\rangle=|i=2\rangle|x=1\rangle$$
$$|4\rangle=|i=2\rangle|x=2\rangle$$
Write out the matrix elements of $$\sigma_2\otimes\eta_1$$

Homework Equations

$$\sigma_a\sigma_b=\delta_{ab} + i\epsilon_{abc}\sigma_c$$

The Attempt at a Solution

I know that that $$\sigma_2\otimes\sigma_1=\begin{bmatrix}0&0&0&-i\\0&0&-i&0\\0&i&0&0\\i&0&0&0\end{bmatrix}$$
And $$\langle i,x| \sigma_2\otimes\eta_1|j,y\rangle = \langle i| \sigma_2|j\rangle \langle x| \eta_1|y\rangle$$
I'm just confused about the matrices being in different spaces, how do I use the defined basis to calculate the matrix elements? I suspect I need the formula given as a relevant equation, but how can I use it with matrices in different spaces.
I'm doing self study with Georgi - Lie Algebras.

 

[e(ho0n3]

Here's what I'm thinking: We have two matrices A and B that represent linear transformations f and g in two spaces U and V with basis (u, u') and (v, v'), respectively. The tensor product space U otimes V will have as a basis (uv, uv', u'v, u'v') and A otimes B will be the matrix representation of f otimes g with the aforementioned basis.

So represenet sigma_2 with respect to (|i=1>, |i=2>) and then eta_1 with respect to (|x=1>, |x=2>) and take their tensor product.

I'm no expert on this though. Caveat emptor.