Fin5al_exam_ _

Fin5al_exam_ _ - MATH 405 Final Examination Fiday 162003...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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 complex-valued 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 (one-to-one), 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 n-dimensional. 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 AMS-TEX 4. Let V be an n-dimensional 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—\/_:_—t2-f(t)g(t)dt. Prove or disprove that 45 is an inner product on the space of continuous complex-valued 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 complex-valued functions on [0,1]. ' ...
View Full Document

This note was uploaded on 09/21/2009 for the course MATH 113 taught by Professor Staff during the Spring '08 term at Maryland.

Page1 / 2

Fin5al_exam_ _ - MATH 405 Final Examination Fiday 162003...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online