Mismatched dimensions in a tensor product with CNOT

[nomadreid]

I am working through an explanation in Nielson and Chuang's Quantum Computation book where they apply a CNOT gate to a state α|0>|00> + β|1>|00>. (The notation here is |0> = the column vector (1,0) and |1>=(0,1), while |00> = |0>|0>, and |a>|b>=|a>⊗|b>, ⊗ being the tensor (outer) product. I am ignoring the constant factor here.) The result is α|0>|00> + β|1>|10>.
But I am working this through via the drudge method, that is, converting everything to old-fashioned matrix formulation, and I run into a problem: the CNOT matrix is a 4x4 matrix, and α|0>|00> + β|1>|00> is a 8 x 1 vector. Using ordinary matrix multiplication, this is a mismatch in dimensions. It appears to me that they are saying that
CNOT (α|0>|00>)= α|0>[CNOT (|00>)]. I am confused.

 

[nomadreid]

I believe I have the missing link now. Instead of purely the CNOT matrix, I need the tensor product of CNOT with the identity (4 x 4) matrix for one of the particles, and the identity tensor product with CNOT for the other one.