Theorem UMCOB says that unitary matrices are characterized as those matrices that carry orthonormal bases to orthonormal
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More precisely, suppose that A is a square matrix of size n and B = {x1, x2, x3, ... , xn} is a basis of Cn. Prove that A is nonsingular if and only if C = {Ax1, Ax2, Ax3, ... , Axn} is a basis of Cn.
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Our first proof relies mostly on definitions of linear independence and spanning which is a good exercise The second proof is shorter and turns on a t...View the full answer
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