# If A is a 3 xx 3-matrix, while x and y are a vector point with x,y,z. Why is the dot product of ⟨A ** x,y⟩ the same as the dot product of ⟨A′ ** y,x⟩. (A′ being the transposed matrix)

If A is a $3×3$-matrix, while x and y are a vector point with x,y,z. Why is the dot product of $⟨{A}^{\ast }x,y⟩$ the same as the dot product of $⟨{A}^{\prime \ast }y,x⟩$
(A′ being the transposed matrix)
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Brooks Hogan
For column vectors $\mathbf{u},\mathbf{v}$, the scalar product is
$⟨\mathbf{u},\mathbf{v}⟩={\mathbf{u}}^{T}\mathbf{v}.$
Therefore
$⟨A\mathbf{x},\mathbf{y}⟩=\left(A\mathbf{x}{\right)}^{T}\mathbf{y}={\mathbf{x}}^{T}{A}^{T}\mathbf{y}=⟨\mathbf{x},{A}^{T}\mathbf{y}⟩=⟨{A}^{T}\mathbf{y},\mathbf{x}⟩.$