Math 110 - Spring 1995 - Bergman - Final

# Math 110 - Spring 1995 - Bergman - Final - THU 19:21 FAX...

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Unformatted text preview: 11/22/2001 THU 19:21 FAX 6434330 MOFFITT LIBRARY 001 George M. Bergman Spring 1995, Math 110, Section 1 15 May, 1995 308 LeConte Hall Final Examination 8: 10—11200 AM 1. (a) (5 points) If V is a ﬁnite—dimensional vector space over a ﬁeld F, and ,6 : {3:1, ,xm} an ordered basis for V, deﬁne what is meant by the coordinate vector [x]ﬁ of an element er with respect to 6. Also indicate, if it is not clear from your deﬁnition, why this coordinate vector is well-deﬁned. (b) (5 points) Suppose V and W are ﬁnite—dimensional vector spaces over a ﬁeld F, with ordered bases [3 = {1:1, ,xm} and y: {311, ,yn}, and that T: V —> W is a linear map. Deﬁne what is meant by [T] g , and give (without proof) a formula for the coordinate vector of the image under T of a vector JCEV in terms of this matrix and the coordinate vector of x. 2. Suppose that S and T are two linear operators on a ﬁnite—dimensional vector space V which commute, i.e., satisfy ST = T8. To avoid confusion, for AG F we will write ETJ for N(T—}L|) and ESJ for N(S—Al). (a) (5 points) Show that for every )LE F, the subSpace ET, A Q V is S—invariant. (b) (10 points) Show that if T is diagonalizable, and if for each eigenvalue )L of T, the operator SETJ (the restriction of S to ET, A) is diagonalizable, then there exists an ordered basis B of V such that [S]B and [T]ﬁ are both diagonal. 3. Let V be a ﬁnite—dimensional inner product Space. For each xEV, let us deﬁne A(x) eV* by A(x)(y) = <y, x> (ye V). (You are not asked to verify that these maps are in fact members of V*.) (a) (12 points) Show that if V is a ﬁnite-dimensional real inner product space, the map A: V ——) V* is linear and invertible. (b) (3 points) Say brieﬂy why the above result fails if V is a complex inner product space. 4. Let V be an inner product space, and W1, W2 nonzero subspaces such that V = W1 6 W2 (i.e., every element of V can be written uniquely as the sum of an element of WI and an element of W2), and such that every element of W1 is orthogonal to every element of W2. Let T1, T2: V —> V be the linear maps deﬁned by T1(x1+x2) = x1, T2(x1 +x2) = 11/22/2001 THU 19:21 FAX 6434330 MOFFITT LIBRARY 002 x2 for x1 6W1, x2 EWZ (the projection maps associated with this direct sum decomposition). (a) (10 points) Show that for all p, :36 F, the operator T = pTl + qT2 has for adjoint the operator T’ = p"T1 + éTz. (b) (10 points) Find and prove necessary and sufﬁcient conditions on p, qE F for the operator T deﬁned in part (a) to be orthogonal (i.e., to satisfy <x, y> 2 <T(x), T(y)> for all x, yEV). 5. Let T be a linear operator on a ﬁnite-dimensional vector space V, let 2L be an eigenvalue of V, and suppose that the generalized eigenspace KR has a Jordan canonical basis {x1,...,x9}, with dot diagram x1 ' x5 ‘ x8 ' x9 x2 x6 13 x7 x4 (a) (7 points) Write out formulas showing the action of T on Kit in terms of this basis. (If you are unsure, it may help to start by writing out formulas for the action of T — ill.) (b) (7 points) Give the matrix for TK/1 (the restriction of T to KA) in terms of the basis {x1,... ,xg}. (You do not have to write out the 0’s, as long as you make it clear what the other matrix entries are, and where they go.) (c) (8 points) Give the dot diagram of a Jordan canonical basis for the restriction of the operator T to the subspace N((T — 3.02) g Kl, naming the basis elements. (d) (8 points) Give the dot diagram of a Jordan canonical basis for the restriction of T to (T * A|)2(K}L), i.e., the subspace {(T - 1020:) | xEKl }, again labeling the basis elements. 6. (10 points) Let T be a linear operator on a ﬁnite—dimensional vector space V, and let W be a T—invariant subspace of V. Show that the minimal polynomial of TW (the restriction of T to W) divides the minimal polynomial of T. ...
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