1. The problem statement, all variables and given/known data Prove that the Unitary matrixes are the only ones that preserve the length of the vectors. 3. The attempt at a solution It's an iff, so I have to prove that a) If the matrix is Unitary, then it preserves the length and b) If the matrix preserves the length, then it's Unitary I could only solve a) (using the canonic inner product for R) Let it be A [tex]\in[/tex] [tex]\Re[/tex]nXn an unitary matrix, x [tex]\in[/tex] [tex]\Re[/tex]n, then (Ax, Ax) = (Ax)T(Ax) = xTATAx = xTx, because ATA = I, for A is unitary, then xTx = (x,x). Then, A preserves the length. But I don't know how to prove b).