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1. We are given a set of three vectors U =[554,035,352-423;-1-53]. The corresponding matrix has rank 3, so U is a basis of a three-dimensional subspace of R5. Using Matlab's orth function, you calculate an orthonormal basis of U: >> o = orth (U) o = -0.6890 0.0867 -0.3858 -0.4110 -0.5032 -0.0045 -0.5313 0.1493 0.0483 -0.1120 -0.6353 0.5875 0.2482 -0.5597 -0.7097 >> a) Verify that o is orthonormal. Hint: this is easy. One matrix multiplication. b) Verify that o and U span the same subspace (so o is an orthonormal basis of span(U)). c) The vector b = [9;14;11;17;-10] lies in the span of U (it is U*[-2;3;1]). We want to write b in the basis O, that is, we want to find x so that Ox = b. We'll do this in two ways: 1. Use linsolve to find X. Verify the answer 2. Use the methods of Fourier coefficients. The two results should be the same. d) The vector C= [-1;2;-4;2;-5] does not line in the span of o (which is the same as the span of U). Find the projection P of c onto o (or U, the same thing). 1. Use linsolve to find X that finds the best solution to Ox = c. With that X, calculate p, the closest point to c in the span of O. 2. Use the methods of Fourier coefficients. Hint: This requires only two matrix multiplications. The two results for p should be the same.

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