Qiskit textbook - exercise 2.2 - why doesn't the correct answer give the same statevector?

Question

In the Qiskit textbook, I'm stuck on one of the exercises in section 2.2. The question is:

Write the state: as two separate qubits.

The answer has already been covered in this question here, and is defined as:

This makes sense to me - we're basically factoring out the first qubit. However, for my own benefit, I tried multiplying out the vectors and I get different results:

Using the original form:

Using the answer supplied:

Why don't these give the same values?

n.b. Apologies for the use of images - I only had MS Word at my disposal and couldn't find an easy way to export the equations bar screen-shotting.

Answer 1

In your answer you are treating $|xy\rangle$ as a direct sum of vectors $|x\rangle$ and $|y\rangle$. However, $|xy\rangle = |x\rangle \otimes |y\rangle$ is actually shorthand for the tensor (Kronecker) product of $|x\rangle$ and $|y\rangle$ (see wiki for more details on the Kronecker product).

In particular if $|x\rangle = \begin{pmatrix} a \\ b\end{pmatrix}$ and $|y\rangle = \begin{pmatrix} c \\ d \end{pmatrix}$ then $$ |xy\rangle = |x\rangle\otimes |y\rangle = \begin{pmatrix} ac \\ ad \\ bc \\ bd \end{pmatrix}. $$