Math 2586 sample Final
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1. Give the detailed deﬁnition of vector spaces.
2. Consider the following 3 ×3 matrix
Find an invertible 3 ×3 matrix Sand a diagonal 3 ×3 matrix Dsuch that
3. Let Abe a 3 ×3 matrix with 3 distinct eigenvalues. Denote by
the characteristic polynomial of A. Show that
4. Determine whether the following matrix is invertible or not:
If Ais invertible, compute A−1.
5. Show that the set of polynomials in xof degree at most 5 is a vector space.
6. Give the deﬁnition of orthogonal matrices.
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.