CNOT Gate

 Controlled NOT gate: 

⏐x, y⟩ → ⏐x, x ⨂ y⟩

Matrix:  

The CNOT maps 

⏐0 0⟩ → ⏐0 0⟩

⏐0 1⟩ → ⏐0 1⟩ 

⏐1 0⟩ → ⏐1 1⟩

⏐1 1⟩ → ⏐1 0⟩

Proof:

⏐00⟩ = [1 0 0 0], ⏐01⟩ = [0 1 0 0], ⏐10⟩ = [0 0 1 0], ⏐11⟩ = [0 0 0 1], all vectors transposed for easy typing. 

Here ⏐0⟩ == [1 0] and ⏐1⟩ == [0 1] and ⏐ab⟩ = ⏐a⟩ ⨂ ⏐b⟩

Apply matrix multiplications: 

CNOT × ⏐00⟩ → ⏐00⟩

CNOT × ⏐01⟩ → ⏐01⟩

CNOT × ⏐10⟩ → ⏐11⟩

CNOT × ⏐11⟩ → ⏐10⟩

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In math, left Kronecker product ⨂ is

For example: