MATH 2270-003 Final Exam Fall 2010Part I1. (5 points) L is a linear transformation from R2to R2such that L 11!= 10!. Whatis L 22!?2. (5 points) Consider the plane V in R4spanned by the vectors ¯v1=1110and ¯v2=1101.Give an orthonormal basis for V .3. (5 points) A = { 32!, 44!} is a basis for R2. Let ¯v = 104!. Write ¯v in A–coordinates.4. (5 points) Are the following vectors linearly independent?123,404,3215. (5 points) Use least squares approximation to find the best fit line for the points (1, 2),(2, 2), (3, 0).6. (5 points) Find all solutions to the following system of equations:2w + 3x + 4y + 5z = 14w + 3x + 8y + 5z = 26w + 3x + 8y + 5z = 117. (5 points) Find an orthogonal matrix Q and diagonal matrix D such that A = QDQTfor A = 3 −1−1 3!.8. (5 points) Show that if B is an invertible matrix and A is similar to B then A isinvertible.9. (5 points) M is an m × n matrix. Prove that the null space of M is a vector subspaceof Rn.10. (5 points) A is an m × n matrix of rank r. Show that the rank of ATA is also r.Conclude that ATA is invertible if and only if the columns of A are linearly independent.Part IIState the Fundamental Theorem of Linear Algebra, Parts 1 and 2, and the Singular ValueDecomposition (SVD) Theorem. Explain, in plain English, what they mean and how theycomplement each other. Give an explicit (actual numbers in the entries) example of a matrixM such that each of the four fundamental subspaces is nontrivial, and such that the twobases in the SVD are not the same. Compute the SVD of M and identify orthonormal basesfor the four fundamental
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