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MATH Dept. Info University of the Philippines Diliman's MATH department has 23 courses in Course Hero with 1564 documents and 4 answered questions.

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    University of the Philippines Diliman
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    Arceo, Unknown, RichardLemence, Dr.Lope, Celeste, Loquias, VonnKeeWong, Dr.Raquel, Paras, ron, Herlo, Walo
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23

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MATH Documents

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  • 123 pages L22_Integrals Involving ln.pdf
    • L22_Integrals Involving ln.pdf
    • University of the Philippines Diliman
    • Introduction to Calculus
    • MATH 100 - Summer 2013
    • Integrals Yielding the Natural Logarithmic Function and Exponential Function Mathematics 100 Institute of Mathematics Math 100 (Inst. of Mathematics) 1 / 16 Recall Recall the derivative of the natural logarithmic function: d 1 (ln x) = . dx x Math 100 (In
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    L22_Integrals Involving ln.pdf

  • 87 pages L23_Area.pdf
    • L23_Area.pdf
    • University of the Philippines Diliman
    • Introduction to Calculus
    • MATH 100 - Summer 2013
    • Area of Plane Regions Mathematics 100 Institute of Mathematics Math 100 (Inst. of Mathematics) Area of Plane Regions 1 / 13 Outline 1 Finding the Area of a Bounded Region 2 Area Between Two Curves Math 100 (Inst. of Mathematics) Area of Plane Regions 2 /
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    L23_Area.pdf

  • 16 pages L25_Applications of Diff Eqtns.pdf
    • L25_Applications of Diff Eqtns.pdf
    • University of the Philippines Diliman
    • Introduction to Calculus
    • MATH 100 - Summer 2013
    • Applications of Differential Equations Mathematics 100 Institute of Mathematics Math 100 (Inst. of Mathematics) Applications of Differential Equations 1 / 16 Outline 1 Obtaining Equations of Curves given the Slope 2 Rectilinear Motion 3 Marginals Math 100
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    L25_Applications of Diff Eqtns.pdf

  • 31 pages lec1.pdf
    • lec1.pdf
    • University of the Philippines Diliman
    • Linear Algebra
    • MATH 114 - Summer 2014
    • Introduction, Forwards and Futures Liuren Wu Options Markets (Hull chapters: 1,2,3,5) c Liuren Wu () Introduction, Forwards & Futures Options Markets 1 / 31 Outline 1 Derivatives 2 Forwards 3 Futures 4 Forward pricing 5 Interest rate parity c Liuren Wu ()
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    lec1.pdf

  • 1 pages 10.1.pdf
    • 10.1.pdf
    • University of the Philippines Diliman
    • Linear Algebra
    • MATH 114 - Summer 2014
    • l Lesson 10 OrthonormalBases and the Gram-Schmidt Process Denitions. Let V be a vector space with inner product ( , ) and 14,11 6 V. - 1, We say u is orthogonal to u (n _L 1) whenever (u, v) : 0, Denitions. An inner roduct on a vector 5 ace V is a functio
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    10.1.pdf

  • 1 pages 9.1.pdf
    • 9.1.pdf
    • University of the Philippines Diliman
    • Linear Algebra
    • MATH 114 - Summer 2014
    • invm mm-m"~# Lesson 9 . I The Fundamental Subspaees 1'1 = 111 1 0 ,. b1 1 0 4r] 7 212 : b2 :5 /l 72 , : b2 :11 4 +12 2 215; + 3ch : ()3 2 3 V 113 2 3 \-/ V A 12 Denition. Let A E MmX"(R). The column space of A, denoted by C(A) consists of all linear combi
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    9.1.pdf

  • 1 pages 7.1.pdf
    • 7.1.pdf
    • University of the Philippines Diliman
    • Linear Algebra
    • MATH 114 - Summer 2014
    • cfw_it [)1 Woh'hwmi ' . dot in '- WWW grow: 8; \/ 91am M 5w . Lesson 7 Linear Independence and Spanning Sets 2. If it happens that V : span S, then S is called the spanning set for V, or V is spanned by S, that is, every vector in V can be expressed as a
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    7.1.pdf

  • 1 pages 9.3.pdf
    • 9.3.pdf
    • University of the Philippines Diliman
    • Linear Algebra
    • MATH 114 - Summer 2014
    • Remark 10. Every matrix A E MWHCR) with full column rank (p(A) : n) has the following properties: 1. There are no free variables in the linear system At : b; 2. All columns of A are pivot columns; 3. MA) = cfw_6; 4, Ar : b has a unique solution or the sys
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    9.3.pdf

  • 1 pages 8.2.pdf
    • 8.2.pdf
    • University of the Philippines Diliman
    • Linear Algebra
    • MATH 114 - Summer 2014
    • 3 ve ofo F5 Example 4. /7 R1 . N 5 Exercises. Do what is asked. 3 L I . . . 1. Is S : cfw_ [ 21] , [g] , [765] linearly independent subset of R2? Nb . / 1. Find a basrs for R3 that Includes the vectors (2,1,1) and (0,3,4). 7 . (L j l cfw_3 Y C T 5 2. Let
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    8.2.pdf

  • 1 pages 6.2.pdf
    • 6.2.pdf
    • University of the Philippines Diliman
    • Linear Algebra
    • MATH 114 - Summer 2014
    • o i l' 2. P2, the set of all polynomials of degree n S 2, is a subspace of 13 3. The set of all symmetric 2 X 2 matrices is a subspace of M2x2(R). ' Remark. We do not need to verify all properties of a vector space to conclude that a subset W of a. ve
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    6.2.pdf

  • 1 pages 9.2.pdf
    • 9.2.pdf
    • University of the Philippines Diliman
    • Linear Algebra
    • MATH 114 - Summer 2014
    • Remark 6. Let M , N be matrices which are compatible for multipication. Then 1. row rank of MN 5 row rank of N and Ktul) : M) 1' . 2. column rank of [MN 3 column rank of M. Theorem. Let A E MmMOR), 1, Q are invertible matrices of app'r'opate sizes. 1. row
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    9.2.pdf

  • 1 pages 7.2.pdf
    • 7.2.pdf
    • University of the Philippines Diliman
    • Linear Algebra
    • MATH 114 - Summer 2014
    • \(oum a1+a3 : 072a1+a2 ~6a3 =0, 3(12 ~ 12% = O. The system is equivalent to 3; a w vaxol tom 1 0 1 a1 0 W ninth ~ We martian WWW u NJMOC m o mu) Example 9. Let V be the set of ordered pairs (:5, y) of real numbers, with operations ($1,111) 69 ($2,212) = (
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    7.2.pdf

  • 1 pages 6.1.pdf
    • 6.1.pdf
    • University of the Philippines Diliman
    • Linear Algebra
    • MATH 114 - Summer 2014
    • . Lesson 6 Vector Spaces and Subspaces Denition. A vector :2 in R", or an nvector is the directed line segment denoted by Ti 502 z:(cci,z2,.,rn)orz: , \vhcrexi,x2,.i,:cnlR. Zn Denition. The norm (or magnitude) of z is Ilzll = 112+ 33; + ~ + x Denition
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    6.1.pdf

  • 1 pages 8.1.pdf
    • 8.1.pdf
    • University of the Philippines Diliman
    • Linear Algebra
    • MATH 114 - Summer 2014
    • \E [4 atoms a V some SFV rm 17v.r~: - . Lesson 8 - Basis and Dimensions Denition. A set of vectors S : cfw_111,712, . . . ,vk in a vector space V is called a basis for V if spans V that is, span S: V and is linearly independent Example 1. 1. Consider the
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    8.1.pdf

  • 1 pages 2.1.pdf
    • 2.1.pdf
    • University of the Philippines Diliman
    • Linear Algebra
    • MATH 114 - Summer 2014
    • i l' r Lesson 2 Solutions of Linear Equations Denitions. . 1. An equation (11:51 + 11212 + + anzn : b, which expresses b in terms of the variables 11:5527 . . ,xn called unknowns and constants a1,a2, . , , ,an is called a linear equation. 2. A s
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    2.1.pdf

  • 1 pages 4.2.pdf
    • 4.2.pdf
    • University of the Philippines Diliman
    • Linear Algebra
    • MATH 114 - Summer 2014
    • 74931 We 1 we 5 1- ,7.4. See Kolmanls book for the proofs of the following properties: Properties of the Determinant 1 2. memes \1 8. 9. . detA : detAT detA det B = det A. ows (or two columns) of A are equal, then det A : 0/ If A has a zero row (or a ze
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    4.2.pdf

  • 1 pages 1.pdf
    • 1.pdf
    • University of the Philippines Diliman
    • Linear Algebra
    • MATH 114 - Summer 2014
    • . Lesson 1 Matrices Properties and Operations Definitions. . 1. An matrix with m rows and TL columns (an m by 77. matrix) A is a rectangular array of numbers: N N (111 (112 1 ~21 (1-12 - ' ' -21: fl = . . i: [afjlmxn = [fl-ijl- L . Ilmi Hm: " ' "mu 'lhc m
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    1.pdf

  • 1 pages 2.2.pdf
    • 2.2.pdf
    • University of the Philippines Diliman
    • Linear Algebra
    • MATH 114 - Summer 2014
    • I A < 3 V T . ' . . 49 a , , "t I]; . 2. Interchange the top row with another row, if necessary, to bring a nonzero entry to the top of the column (111 (112 a1" aCI b1 found in Step 1. . I 6121 (122 02" $2 12 I I . Denition. Let A : _ . . II : , Ib : .
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    2.2.pdf

  • 1 pages 5.1.pdf
    • 5.1.pdf
    • University of the Philippines Diliman
    • Linear Algebra
    • MATH 114 - Summer 2014
    • Lesson 5 Cofactor Expansion and Applications of the Determinant 1 1 0 Example 3. Given A = 2 3 4 . Let's [ind rst all of its nine col'actors. Denition. If A is a square matrix, then the (i,j)-minor of A denoted by MUM) (or simply Miil is defined to be 5 8
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    5.1.pdf

  • 1 pages 3.2.pdf
    • 3.2.pdf
    • University of the Philippines Diliman
    • Linear Algebra
    • MATH 114 - Summer 2014
    • Therefore, 1 2 o 3 1 o 0 2 e A Method for Finding the Inverse of a Nonsingular Matrix 1. Form the n X 271 augmented matrix [A : In]. 4:1 1 3 :> ~1 ~5 10 5 O 1 0 0 0 A? 1 0 1. 0 O 1 0 1 O 3 2 0 2 *2 ~3 2. Using elementary row operations, transform this mat
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    3.2.pdf

  • 1 pages 2.3.pdf
    • 2.3.pdf
    • University of the Philippines Diliman
    • Linear Algebra
    • MATH 114 - Summer 2014
    • . m wt 1 I 11 [zlr7KL'C/(KS Example 8. Consider the homogeneous system 5 Find the value(s) of c for which the system l , m+zg~3z3+2x4:0 1 1 73 2 ' avey = 1 C/ l 2x1 2m + 2:53 2 0 a 2 72 2 0 (coefcient matrix) SOV; Fj 01' W y = c I C N . v1+c2~13=0 A1
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    2.3.pdf

  • 1 pages 3.1.pdf
    • 3.1.pdf
    • University of the Philippines Diliman
    • Linear Algebra
    • MATH 114 - Summer 2014
    • a: Lesson 3 Inverse of a Matrix Denitions. 1. An n X n matrix A is said to be nonsingular or invertible if there exists a unique n X n matrix B such that AB = BA ; I". The matrix B is called the inverse of A and denote it b i 2. If no such matrix B exists
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    3.1.pdf

  • 1 pages 5.2.pdf
    • 5.2.pdf
    • University of the Philippines Diliman
    • Linear Algebra
    • MATH 114 - Summer 2014
    • i. wmnuW ~ .- Cramers Rule Theorem, 1f Ax : b is a system of n linear' equations in n unknowns where .r : detA 7 0, then the system has a unique solution given by _ dot. A: :t' I detA ' where A, is the matrix obtained by replacing the entries in th
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    5.2.pdf

  • 1 pages 4.1.pdf
    • 4.1.pdf
    • University of the Philippines Diliman
    • Linear Algebra
    • MATH 114 - Summer 2014
    • Lesson 4 Denition and Properties of Determinants One application of determinants 18 to determine whether a linear system with n equations in n unknowns has a unique solution We rst dene the concept of permutations before dening the detein),.inants Denitio
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    4.1.pdf

  • 1 pages RANKING
    • RANKING
    • University of the Philippines Diliman
    • Basic Math
    • MATH 1 - Spring 2012
    • Actually Isis, based on my research, the effects of student rankings in college can be bad or good depending on a students performance. The possible advantages could be 1) upon seeing his or her standing, a student can be motivated to perform better to su
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    RANKING

  • 4 pages Supplementary Notes For Surface Integrals
    • Supplementary Notes For Surface Integrals
    • University of the Philippines Diliman
    • Elementary Analysis III
    • MATH 55 - Winter 2010
    • SSX%AG- 6 5, X'fG-(i'7 Am a (d Gala/Ff TGECM] Sow-Hon 11 :7 0 1n mrpw, W be. 01st M 3 576") = cfw_X I-xz,i>, mueb g %: IX' @,\j)e'D EX I<".)0> 9 Q4 (cm) 1) D Boundwxcs: (allelic) 3>l-) M w - a 5 c are [0. q 38:738. 24:93, l-K-ld> (19,3090 '3"qu = <I,o,-I
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    Supplementary Notes For Surface Integrals

  • 8 pages Exercise 02 - Long Exam 02
    • Exercise 02 - Long Exam 02
    • University of the Philippines Diliman
    • Elementary Analysis III
    • MATH 55 - Winter 2010
    • I. Triple Integrals 1. Evaluate the following triple integrals. ZZZ (a) xyz 2 dV , where S = [0, 1] [1, 2] [0, 3] Z SZ Z (2x + y sin z) dV , where G = [1, 1] [0, 2] 0, 2 (b) ZG ZZ (c) xy sin yz dV , where S is bounded by the coordinate planes, x = , y = 2
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    Exercise 02 - Long Exam 02

  • 3 pages Notes 02 - Long Exam 02 - Answer Key
    • Notes 02 - Long Exam 02 - Answer Key
    • University of the Philippines Diliman
    • Elementary Analysis III
    • MATH 55 - Winter 2010
    • Math 55 EXAM 2 October 22, 2016 Show all necessary solutions and box your nal answers. Use only black or blue non-erasable ink in writing. Do not detach any page from your bluebooks. Follow directions carefully. 1. (8 pts.) Let 13(2, y, z) = (22:31, y3, :
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    Notes 02 - Long Exam 02 - Answer Key

  • 3 pages Notes 03 - Long Exam 03 - Answer Key
    • Notes 03 - Long Exam 03 - Answer Key
    • University of the Philippines Diliman
    • Elementary Analysis III
    • MATH 55 - Winter 2010
    • MATHEMATICS 55 Third Examination 03 December 2016 Write legibly and box every nal answer. Use black or blue non-erasable ink only. Show all necessary . solutions for parts I-VII. . 3" . I. Consider the sequence cfw_ (2n)! n=1. (4 paints) 1. Show that th
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    Notes 03 - Long Exam 03 - Answer Key

  • 5 pages Exercise 03 - Long Exam 03
    • Exercise 03 - Long Exam 03
    • University of the Philippines Diliman
    • Elementary Analysis III
    • MATH 55 - Winter 2010
    • I. Sequences 1. Find the first five terms of the following sequences. 1 n (a) (d) n 2n + 1 n=1 n=1 1 (1)n (n + 1) (b) 2 (e) n n=1 3n n=0 n (c) (f) n 3 n=3 n + 1 n=1 (k) cfw_f (n)n=1 , where f (n) = 1 when n is odd, f (n) = n n o cos 6 n n=0 e (h) n
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    Exercise 03 - Long Exam 03

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