Math 2568 m2568au17final_sample

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Published on 31 Jan 2019
School
Ohio State University
Department
Mathematics
Course
MATH 2568
Professor
Math 2586 sample Final
Hsian-Hua Tseng
December 2017
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Problems
1. Give the detailed definition of vector spaces.
2. Consider the following 3 ×3 matrix
A=
111
121
112
.
Find an invertible 3 ×3 matrix Sand a diagonal 3 ×3 matrix Dsuch that
AS =SD.
3. Let Abe a 3 ×3 matrix with 3 distinct eigenvalues. Denote by
c0+c1x+c2x2+c3x3
the characteristic polynomial of A. Show that
c0I3+c1A+c2A2+c3A3=O.
4. Determine whether the following matrix is invertible or not:
A=
111
121
112
.
If Ais invertible, compute A1.
5. Show that the set of polynomials in xof degree at most 5 is a vector space.
6. Give the denition of orthogonal matrices.
1
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Document Summary

Problems: give the detailed de nition of vector spaces, consider the following 3 3 matrix. Find an invertible 3 3 matrix s and a diagonal 3 3 matrix d such that. As = sd: let a be a 3 3 matrix with 3 distinct eigenvalues. Denote by c0 + c1x + c2x2 + c3x3 the characteristic polynomial of a. Show that c0i3 + c1a + c2a2 + c3a3 = o: determine whether the following matrix is invertible or not: If a is invertible, compute a 1: show that the set of polynomials in x of degree at most 5 is a vector space, give the de nition of orthogonal matrices.