MATH V2010 Columbia S11Final LinearAlgebra S11

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Published on 31 Jan 2019
School
Columbia University
Department
Mathematics
Course
MATH V2010
Professor
Final Exam
Linear Algebra, Dave Bayer, May 10, 2011
Name:
[1] (5 pts) [2] (5 pts) [3] (5 pts) [4] (5 pts) [5] (5 pts) [6] (5 pts) [7] (5 pts) [8] (5 pts) TOTAL
Please work only one problem per page, starting with the pages provided. Clearly label your
answer. If a problem continues on a new page, clearly state this fact on both the old and the new
pages.
[1] By least squares, find the equation of the form y=ax +bwhich best fits the data
x1y1
x2y2
x3y3
=
0 1
1 1
3 2
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[2] Extend the vector (1,1,1,2)to an orthogonal basis for R4.
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[3] Find the orthogonal projection of the vector (1,0,0,0)onto the subspace of R4spanned by the
vectors (1,1,1,0)and (0,1,1,1).
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Document Summary

[1] (5 pts) [2] (5 pts) [3] (5 pts) [4] (5 pts) [5] (5 pts) [6] (5 pts) [7] (5 pts) [8] (5 pts) total. Please work only one problem per page, starting with the pages provided. If a problem continues on a new page, clearly state this fact on both the old and the new pages. [1] by least squares, nd the equation of the form y = ax + b which best ts the data x1 y1 x2 y2 x3 y3. [2] extend the vector (1,1,1,2) to an orthogonal basis for r4. [3] find the orthogonal projection of the vector (1,0,0,0) onto the subspace of r4 spanned by the vectors (1,1,1,0) and (0,1,1,1). [4] find the matrix a which projects r4 orthogonally onto the subspace spanned by the vectors (1,1,1,1) and (1,1,2,2). [5] find the eigenvalues and corresponding eigenvectors of the matrix. [6] find the matrix exponential eat, for the matrix.