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Problem 3:
(A)Let ๐ด=[โ102๐1+๐๐โ2๐00๐].
[3 Marks] i)Determine the two eigenvalues of ๐ด.
[2 Marks] ii)Show that ๐ฃ=[โ110] is an eigenvector of ๐ด.
[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 and ๐ท such that ๐ด=๐๐ท๐โ1.
[1 Mark] v)Explain how you would find ๐ด80 using the diagonalization (do not calculate ๐ด80).
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Problem 5:
(A)Let ๐={[ โ10101],[11110],[ โ1โ102โ1]} and let ๐=๐๐๐๐ ๐.
[2 Marks] i)Show ๐ is an orthogonal set (under the standard dot product).
[5 Marks] ii)Provide the orthogonal decomposition of ๐ฆ=[0475โ1] in terms of ๐ฆฬโ๐ and ๐งโ๐โฅ.
[1 Mark] iii)๐โฅ can be found by finding the Null space of which matrix?
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Problem 7:
(A)Consider the following set of data points: {(โ2,1),(โ1,2),(0,1),(1,5),(2,6)}.
[2 Marks] i)Construct a matrix ๐ด and column vector ๐ such that ๐ด๐ฃ=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?
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Problem 8:
(A)Consider an inner product on ๐22(๐น) given by ใ๐ด,๐ตใ=๐๐๐๐๐(๐ด๐ต๐) for the following:
[2 Marks] i)Show that ๐ต={[1001],[โ11โ11]} is an orthogonal set.
[4 Marks] ii)Let ๐=๐ ๐๐๐(๐ต), find ๐๐๐๐๐(๐ฆ) where ๐ฆ=[1โ463].
[4 Marks] iii)Find the point in ๐ closest to ๐ฆ and the shortest distance from ๐ฆ to ๐.
[1 Mark] iv)Explain how the dimensions of ๐ and ๐โฅ are related.
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Problem 9:
[1 Mark] (A)Let ๐(๐ฅ)=๐ฅ12+2๐ฅ22+2๐ฅ1๐ฅ2โ4๐ฅ2๐ฅ3 be a quadratic form from ๐น3โ๐น. Find thematrix of the quadratic form.
(B)Consider a quadratic ๐(๐ฅ) with matrix ๐ด=[311131113].
[9 Marks] i)Find a change of variable from ๐ฅ to ๐ฆ and the corresponding quadratic form ๐
(๐ฆ) thatresults in ๐
(๐ฆ) not having cross-product terms. (๐๐๐ก(๐ดโ๐๐ผ)=โ(๐โ2)2(๐โ5))
[2 Marks] ii)Is ๐(๐ฅ) positive definite, positive semidefinite, negative definite, negative semidefinite,
or indefinite? Explain how you know.

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