**Unformatted text preview: **03/23/2001 FRI 17:42 FAX 6434330 MOFFITT LIBRARY 001 Math 110 - Final Exam
Spring 2000 — Nate Brown 1) (10pts) Assume dim(V) : 6, U C V is a subspace and dim(U) = 4.
Prove that there exist one dimensional subspaces U1, U2 C V such that V =
U EB U1 33 Hg. 2) (10pts) Assume T E C(V) is invertible and {v1,... ,vk} C V is a lin-
early independent set of vectors. Prove that {T(n1), . . . ,T(-uk)} is also linearly
independent . 3) (lOpts) Let < -,- > be the dot product on C2 (i.e. < (331,311), (3:2,y2) >:
3215+ 3,123) and T E £(C2) be deﬁned by T(:r,y) = + 215$ +2311). Write
down a formula for the adjoint of T with respect to < -, - >. (It does not sufﬁce
to just write down some matrix!) 4) (10pts) Assume T E [3032013) has minimal polynomial (z * 1)(z + 4)?
Find the matrix of T in Jordan form. 5) Let U, V be vector spaces with di-rri(U) > d‘im(V). a) (lOpts) Prove that if T E £(U, V) then T is not injective. b) (10pts) Construct some T E £(U, V) which is surjectivei a, b # ia+3c —b+u!
6) DeﬁneTE£(M2(C)) byT( c d)_( ic 2d a) (10pts) Find all the eigenvalues of T. (Hint: Consider the matrix of T
with respect to the canonical basis.) b) (10pts) Compute the characteristic polynomial of T. c) (lOpts) What is the dimension of the generalized eigenspacc corresponding
to the eigenvalue i? d) (10pts) Find det(T). 7)a) (lOpts) Construct an operator T E £(M2 ((3)) whose minimal polyno-
mial is z(z w 3)2 and generalized eigenspace corresponding to 0 is one dimen—
sional. (Cive both a formula for the operator and it’s matrix in Jordan form.) b) (10pts) Construct an operator 5' E £(M;(<C)) whose minimal polynomial
is 2(2 7 3)2 and generalized eigenspace corresponding to O is two dimensional.
(Give both a formula for the operator and it’s matrix in Jordan form.) 03/23/2001 FRI 17:42 FAX 6434330 MOFFITT LIBRARY 002 8) For each A 6 IF deﬁne TA 6 £(Pm(IF),lF] by Tim) 21200. a) (Spts) Prove that ker(T,\) = ker(TS) if and only if A = X. (Hint: consider
the polynornials 107(2) : z — 7.] b) (lets) Prove that dim(ker(TA)) 2 m. c) (153th Let U C PmUF) be the one dimensional subspace spanned by
the polynomial 10(2) 2 2. Prove that for every nonzero A E IF there exists a
Subspace UA c 79mm) such that 2') U,\ = US if and only ifA = Z\ and a) 73mm =
U EB U ,\ for each (nonzero) A. 9) Let Bl : {(1,0,0,0}, (0,1,0,0), (0,0,1,0), (0,0,0,1)} be the canonical
basis ofC‘, 32 = {(1,0,0,0), (1,1,0,0), (0, 0,1,0), (u,0,1,1)} and T E 130134)
be deﬁned by Tm, b,c,d) = ((1 —— as + (21' + 1)b, —z‘a + (i + 2)b,2c,c). a) (Spins) Compute M(T,Bl, Bl).
b) (10pts) Compute the two change of basis matrices M (I , Bl, Ba) and M(Ii B2; B1)'
1 i 0 0
“i 2 0 0
c) (opts) Use part b) to show that M(T, 32,132) = 0 0 1 1
0 O 1 1 d) (lOpts) Prove that T is not normal with reapect the canonical dot product
on CC“. e) (10pts) Prove that there exists a basis of (0“ consisting of eigenvectors of
T. f) (10pts) Construct an inner product on (C‘1 such that T is normal with
respect to that inner product. ...

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