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for such a matrix.    Answer:   (1) First we calculate $C$. Let $Cj$ $C_j$  denote the $j$th column of C. Since C has orthonormal columns, each $Cj$ $C_j$  has norm 1. Then  $$||C|| = \sqrt{\mathop{\sum_{i=1}^n\sum_{j=1}^n}C_{ij}^2}$$  $$=\sqrt{\mathop{\sum_{j=1}^n(\sum_{i=1}^n}C_{ij}^2)}$$ 
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Now we can put the pieces together:  $$||CX||^2 = ||x_1C_1 + x_2C_2 + · · · + x_nC_n||^2$$  = ||X||2. $$= ||X||2.$$  Since norms are nonnegative, we can conclude that $||CX|| = ||X||$.