Linear Algebra Problems

4 Problem 3: (A)Let 𝐴=[−102𝑖1+𝑖𝑖−2𝑖00𝑖]. [3 Marks] i)Determine the two eigenvalues of 𝐴. [2 Marks] ii)Show that 𝑣=[−110] is an eigenvector of 𝐴. [4 Marks] iii)Find the eigenbasis for the other eigenvalue (the one not found in ii). [2 Marks] iv)Diagonalize the matrix by finding 𝑃 and 𝐷 such that 𝐴=𝑃𝐷𝑃−1. [1 Mark] v)Explain how you would find 𝐴80 using the diagonalization (do not calculate 𝐴80). 6 Problem 5: (A)Let 𝑈={[ −10101],[11110],[ −1−102−1]} and let 𝑊=𝑆𝑝𝑎𝑛 𝑈. [2 Marks] i)Show 𝑈 is an orthogonal set (under the standard dot product). [5 Marks] ii)Provide the orthogonal decomposition of 𝑦=[0475−1] in terms of 𝑦̂∈𝑊 and 𝑧∈𝑊⊥. [1 Mark] iii)𝑊⊥ can be found by finding the Null space of which matrix? 8 Problem 7: (A)Consider the following set of data points: {(−2,1),(−1,2),(0,1),(1,5),(2,6)}. [2 Marks] i)Construct a matrix 𝐴 and column vector 𝑏 such that 𝐴𝑣=𝑏 could be used to solvethe least squares line. [5 Marks] ii)Solve the least squares line that you found in i) and state the equation of the line. [2 Marks] iii)Find the residual vector for the least squares solution you found in ii). [2 Marks] iv)If instead we want to find the least squares cubic, what would be the matrix A? 9 Problem 8: (A)Consider an inner product on 𝑀22(𝑹) given by 〈𝐴,𝐵〉=𝑇𝑟𝑎𝑐𝑒(𝐴𝐵𝑇) for the following: [2 Marks] i)Show that 𝐵={[1001],[−11−11]} is an orthogonal set. [4 Marks] ii)Let 𝑊=𝑠𝑝𝑎𝑛(𝐵), find 𝑝𝑟𝑜𝑗𝑊(𝑦) where 𝑦=[1−463]. [4 Marks] iii)Find the point in 𝑊 closest to 𝑦 and the shortest distance from 𝑦 to 𝑊. [1 Mark] iv)Explain how the dimensions of 𝑊 and 𝑊⊥ are related. 10 Problem 9: [1 Mark] (A)Let 𝑃(𝑥)=𝑥12+2𝑥22+2𝑥1𝑥2−4𝑥2𝑥3 be a quadratic form from 𝑹3→𝑹. Find thematrix of the quadratic form. (B)Consider a quadratic 𝑄(𝑥) with matrix 𝐴=[311131113]. [9 Marks] i)Find a change of variable from 𝑥 to 𝑦 and the corresponding quadratic form 𝑅(𝑦) thatresults in 𝑅(𝑦) not having cross-product terms. (𝑑𝑒𝑡(𝐴−𝜆𝐼)=−(𝜆−2)2(𝜆−5)) [2 Marks] ii)Is 𝑄(𝑥) positive definite, positive semidefinite, negative definite, negative semidefinite, or indefinite? Explain how you know.