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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.