# Exercise 9 8 points find all least squares solutions

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Exercise 9 (8 points):Find all least squares solutions of the following system ofequations:X+Y+ 2Z= 22X+ 2Y+ 3Z= 13X+ 3Y+ 4Z= 3Solution:
Exercise 10 (10 points):LetC([-2,2]) denote the vector space of continuous functionsf: [-2,2]!R, equipped with the inner product:hf, gi=Z2-2f(t)g(t)dt(1) Consider the functionf(t) =t. Compute||f||.(2) Construct an orthonormal basis (with respect to the inner producth-,-iabove)of the sub-spaceP1(R) of polynomials of degree1.(3) Consider the functiong(t) =t3.Compute ProjP1(R)(g), and draw it (togetherwithg(t)) on the interval [-2,2].Solution:
Exercise 11 (8 points):Compute the determinant of the following matrix (show yourwork):A=0BBBB@10000122221030010344103051CCCCASolution:
Exercise 12 (12 points):Consider the following matrix:A=266401111011110111103775(1) Write down the characteristic polynomialfA(X). What are the real eigenvaluesofA, and their corresponding algebraic multiplicities?(2)Ais diagonalizable. Find a basis ofR4consisting of eigenvectors ofA.(3) Find an orthonormal basis ofR4consisting of eigenvectors ofA.(4) Use your answer to computeA7by diagonalizingA.Solution: