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4 Problem 3: (A)Let ๐ด=[โˆ’102๐‘–1+๐‘–๐‘–โˆ’2๐‘–00๐‘–]. [3 Marks] i)Determine the two eigenvalues of ๐ด. [2 Marks] ii)Show that ๐‘ฃ=[โˆ’110] is an eigenvector of ๐ด. [4 Marks] iii)Find the eigenbasis for the other eigenvalue (the one not found in ii). [2 Marks] iv)Diagonalize the matrix by finding ๐‘ƒ and ๐ท such that ๐ด=๐‘ƒ๐ท๐‘ƒโˆ’1. [1 Mark] v)Explain how you would find ๐ด80 using the diagonalization (do not calculate ๐ด80). 6 Problem 5: (A)Let ๐‘ˆ={[ โˆ’10101],[11110],[ โˆ’1โˆ’102โˆ’1]} and let ๐‘Š=๐‘†๐‘๐‘Ž๐‘› ๐‘ˆ. [2 Marks] i)Show ๐‘ˆ is an orthogonal set (under the standard dot product). [5 Marks] ii)Provide the orthogonal decomposition of ๐‘ฆ=[0475โˆ’1] in terms of ๐‘ฆฬ‚โˆˆ๐‘Š and ๐‘งโˆˆ๐‘ŠโŠฅ. [1 Mark] iii)๐‘ŠโŠฅ can be found by finding the Null space of which matrix? 8 Problem 7: (A)Consider the following set of data points: {(โˆ’2,1),(โˆ’1,2),(0,1),(1,5),(2,6)}. [2 Marks] i)Construct a matrix ๐ด and column vector ๐‘ such that ๐ด๐‘ฃ=๐‘ 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 ๐‘€22(๐‘น) given by ใ€ˆ๐ด,๐ตใ€‰=๐‘‡๐‘Ÿ๐‘Ž๐‘๐‘’(๐ด๐ต๐‘‡) for the following: [2 Marks] i)Show that ๐ต={[1001],[โˆ’11โˆ’11]} is an orthogonal set. [4 Marks] ii)Let ๐‘Š=๐‘ ๐‘๐‘Ž๐‘›(๐ต), find ๐‘๐‘Ÿ๐‘œ๐‘—๐‘Š(๐‘ฆ) where ๐‘ฆ=[1โˆ’463]. [4 Marks] iii)Find the point in ๐‘Š closest to ๐‘ฆ and the shortest distance from ๐‘ฆ to ๐‘Š. [1 Mark] iv)Explain how the dimensions of ๐‘Š and ๐‘ŠโŠฅ are related. 10 Problem 9: [1 Mark] (A)Let ๐‘ƒ(๐‘ฅ)=๐‘ฅ12+2๐‘ฅ22+2๐‘ฅ1๐‘ฅ2โˆ’4๐‘ฅ2๐‘ฅ3 be a quadratic form from ๐‘น3โ†’๐‘น. Find thematrix of the quadratic form. (B)Consider a quadratic ๐‘„(๐‘ฅ) with matrix ๐ด=[311131113]. [9 Marks] i)Find a change of variable from ๐‘ฅ to ๐‘ฆ and the corresponding quadratic form ๐‘…(๐‘ฆ) thatresults in ๐‘…(๐‘ฆ) not having cross-product terms. (๐‘‘๐‘’๐‘ก(๐ดโˆ’๐œ†๐ผ)=โˆ’(๐œ†โˆ’2)2(๐œ†โˆ’5)) [2 Marks] ii)Is ๐‘„(๐‘ฅ) positive definite, positive semidefinite, negative definite, negative semidefinite, or indefinite? Explain how you know.

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Linear Algebra Problems
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