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Unformatted text preview: MATH 405 Final Examination Fiday 162003, 8:00—10:00 am. Instructions. Solve any six of the following seven problems, which have equal weight. On the
answer sheet for the problem to be omitted, write “Omit”. Carefully show all yours steps, justify
all your assertions, state precisely any definitions and theorems that you use, and explain your
arguments in compete English sentences. Cross out any material that is not to be graded. If you
use more than one answer sheet for a given problem, indicate that fact on each answer book. You must submit an answer sheet (possibly blank) for each problem. You may not use calculators,
notes, or texts. In accord with university policy, you are requested to handwrite and sign the following pledge on your ﬁrst answer sheet: I pledge on my honor that I have neither given nor received any
unauthorized assistance on this ezaminatz’on. 1. (a) Show that the functions t +——) cu“ =2 fk(t), k = 0,1,...,n, are linearly independent
vectors on the space of continuous complexvalued functions on [0, 273']. (You should recall that
e” = c050 + isin 9, but you would be best served by using the exponential form.) (b) On the subspace S := span {fo, . . . , fn} deﬁne the linear transformation A by (A f)(t) := 1 + 2% f (Note that 1 = e0.) Find the components of the matrix of A with respect to the basis
{fo, . . . , fn} and actually display the matrix. 2. (a) Let f be any function from a set .A to 8. Deﬁne what it means for f to be surjective (onto),
to be injective (onetoone), and to be invertible. (b) Let V be the vector space of inﬁnite sequences of complex numbers of the form k=1 On V define the three linear transformations A, B, C’ by A$Z=(01€1)€2:€31 ‘ ' )1
Bx := (£2,€3,€4a~~): £1 £2 £3
0131: . Determine whether each of these linear transformations is surjective or not, injective or not, and
invertible or not. (For treating C', it might be useful to recall that 2:11 < 3. Let m,y, . .. be vectors in an inner product space ’H. A special linear transformation from ’H
to itself denoted a: (8) y called a tensor product is deﬁned by (21 ® y)z 1= $(y, 2) E (11,2)03 (a) A linear transformation from ’H to itself may be speciﬁed by what it does to a basis. Let {a2}
be a basis (not necessarily orthonormal) for H, assumed to be ndimensional. Express the linear
transformation that takes of, . . . , a; respectively into a prescribed set of vectors b1, . . . , bn as a
sum of n tensor products. Hint: Incorporate the basis {at} dual to {a2} into the tensor products. (b) Let {ck} and {dk} be bases for 3%. Again using that a linear transformation from ’H to itself
may be speciﬁed by what it does to any (judiciously chosen) basis, show that {61C (8 dl} is a basis
for the vector space £(’H) of linear transformations from ’H to itself. Typeset by AMSTEX 4. Let V be an ndimensional vectors space. Let A E £(V) have distinct eigenvalues A1, . . . ,An
with corresponding eigenvectors ul, . . . , un. Let p be a given scalar and let f E 213:1 <pkuk be a
given vector. For each such p and f determine whether the equation Ax — pa: = f
has a solution cc 2 22:1 ékuk, and if it does, ﬁnd the most general such solution. 5. Find the eigenvalues, eigenvectors, generalized eigenvectors of index 2, and generalized eigen~
vectors of index 3 for the matrix operator A from C3 to itself: 4 —1 O
A = 3 1 —1
1 0 1
Hint: An eigenvalue of A is 2.
6. Let 81, . . . , en be an orthonormal set in an inner product space ’H. Show that the following statements (1)*(5) are equivalent by proving that (1) => (2), (2) :> (3), (3) => (4), (4) :> (5),
(5) => (1) (1) If :1: is in H, then = 22:1(m,ek)2. (2) Ifa: is in’H and if(x,ek) =0 for k: 1,...,n, then x=0. (3) span {el,...,en} = ’H. (4) Ifa: is in ’H, then a; = Z£:I(m,ek)ek. (5) If :r and y are in 7%, then (2:,y) = 2221(23, ek)(ek,y). ( 7. (a) Define 1
45mg) :2/0 —1—\/_:_—t2f(t)g(t)dt. Prove or disprove that 45 is an inner product on the space of continuous complexvalued functions
on [0, 1]. (b) Define 1
5W, 9) := f(t)§(t) dt. 1/2
Prove or disprove that W is an inner product on the space of continuous complex—valued functions
on [0,1]. (G) Let f and g be n—tuples (f1, . . . , fn) and (91,. . . ,9”) of continuous complex~valued functions
on [0,1]. Define 1 n .
9mg) r==/O ka(t)gk(t)dt.
k=1 Prove or disprove that 9 is an inner product on the space of n—tuples of continuous complexvalued
functions on [0,1]. ' ...
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This note was uploaded on 09/21/2009 for the course MATH 113 taught by Professor Staff during the Spring '08 term at Maryland.
 Spring '08
 staff

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