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)
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)).
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
The two results for p should be the same.
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