5. Orthongonalization and least squares [2+3+3pt]. (a) Given any two nonzero vectors x and y in R^n, construct a Householder matrix H, such that Hx is a scalar multiple of

y. Is the matrix H unique? (b) Use Householder matrices to compute the QR-factorization of the matrix: (c) We believe that a real number Y is approximately determined by X with the model function Y = a exp(X)+bX² + cX + d . We are given the following table of data for the values of X and Y: Using the above data points, write down 7 equations in the four unknowns a, b, c, d. The least squares solution to this system is the best fit function. Write down the normal equations for this system, solve them in MATLAB. Plot the data points (X,Y) as points and the best fit function.

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