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The final exam for the course Math 2210 - Linear Algebra. The exam was held on May 14, 2014, in Classroom URHG01. The exam consists of 6 problems on 6 pages, worth a total of 100 points. The instructions for the exam include showing all work and justifying answers. Academic integrity is expected of all Cornell University students at all times. The exam covers topics such as eigenvalues, diagonalization, least-square line, linear transformations, and inner products.
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Academic integrity is expected of all Cornell University students at all times, whether in the presence or absence of members of the faculty. Understanding this, I declare I shall not give, use, or receive unauthorized aid in this examination. Please sign below to indicate that you have read and agree to these instructions.
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. Eigenvalues of A are 1, 1 , 4.
(1). Orthogonally diagonalize the matrix A. (2). Does there exist a unit vector x in R^3 such that xT^ Ax = 0? Does there exist a unit vector x in R^3 such that xT^ Ax = 6? If yes, find the vector. If no, explain the reason. (3). Suppose
A is also symmetric, compute
Hint: Consider the orthogonal diagonalization of
A, then compute its square, and compare with the orthogonal diagonalization of A.
〈f, g〉 =
0
[f (x)g(x) + f ′(x)g′(x)]dx
(2). For any two vectors x, y ∈ Rn,
‖x + y‖^2 + ‖x − y‖^2 = 2‖x‖^2 + 2‖y‖^2.
(3). Let U be an n × n orthogonal matrix, then any real eigenvalue of U must be 1. (4). If two symmetric matrices A, B ∈ Mn×n are both positive definite, then A + B is also positive definite.
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