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and that / + fA + gA + 18 is the characteristic polynomial of A, where f and g are some real numbers, prove that A is diagonalizable. 7. Let S = (up,...,u,) be a set of column vectors. Suppose S is an orthonormal basis for a subspace V of R". For a vector v E V, (v); denotes the coordinates of v relative to S. Let A be an m x n matrix. Define M = |/Au,ll + . . . + [|Auell. (a) (3 points) Prove that for any ve V, if (v)s = (C1, . . ., C.), then llvil s Icjl + . .. + [cil. (b) (4 points) Prove that for any ve V, if (v)s = (c1, . . .,c.) and |/vl|