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    Column Space of Matrix A and ref(A)

    Homework Statement Given a matrix A. So I can reduce A to ref(A). Let's say in ref(A), the columns that contain leading ones are column 1, 3, and 5. True or false: (a) Columns 1, 3, and 5 from ref(A) form the column space of ref(A). (b) The corresponding column 1, 3, and 5 from the original...
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    Derive the Integrating Factor for Homogeneous DE

    Homework Statement I have this statement: If M(x,y)dx+N(x,y)dy=0 is a homogeneous DE, then μ(x,y)=\frac{1}{xM+yN} is its integrating factor. The problem is, how do we derive this integrating factor? Homework Equations For homogeneous DE, we have f(kx,ky)=k^n*f(x,y) We also have...
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    Distance to the corner of a rectangle

    Homework Statement This question is taken from 2011 Malaysian Mathematical Olympiad. Mary is standing in a rectangular garden. Her distance to the four corners of the garden are 6 m, 7 m, 9 m and d m, respectively, where d is an integer. Find d. Homework Equations Triangle...
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    Find the limit

    Homework Statement Find the lim of \frac{\sqrt{6-x}-2}{\sqrt{3-x}-1}when x approaches 2, without using L'hopital rule or the definition. Homework Equations The Attempt at a Solution I try to multiply both the numerator and the denominator to get \frac{(\sqrt{6-x}-2)(\sqrt{3-x}-1)}{2-x}...
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    System of Linear Equations - Proving

    Homework Statement Given that the values for a, b, c, d, e and f for the system ax+by=e, cx+dy=f has two different solutions. Show that ax+by=0, cx+dy=0 also has two different solutions. Homework Equations The Attempt at a Solution There're three cases of how two straight lines...
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    MATLAB Neural Network Classification Problem

    Homework Statement P = [data(1:65,11:100) data(1:65,411:500)]; % Input, all together 180 data set, 65 dimension T = [ones(1,90) zeros(1,90); zeros(1,90) ones(1,90)]; % Actual output, first 90 data set belong to class I and the next 90 belong to class II net =...
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    Differences between Two Notations in DE

    Homework Statement Is there any difference between the two notations below? Or can they be used interchangeably? \tilde{x} and \underline{x} (not an underline, but the tilde is at the bottom of the unknown x) Homework Equations The Attempt at a Solution From my observation...
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    First Order ODE

    Homework Statement Solve y'=x^2+y^2 with initial condition y(0)=1. Homework Equations This is a first order ODE. The Attempt at a Solution I have tried separable variable, exact, and homogeneous and non-homogeneous, but none of them work. It's neither linear nor Bernoulli...
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    Linear Independence and Span

    Homework Statement You are given 4 vectors in R^4 which are linearly independent. Do they always span R^4? Homework Equations The Attempt at a Solution Intuitively, I think the answer is yes. I know if I want to show they span R^4, I need to use the general terms, but all I can think of...
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    Fourier Transform and Dirac Delta Function

    Homework Statement I am new to FT and dirac delta function. Given the following signal: x\left(t\right)=cos\left(2\pi5t\right)+cos\left(2\pi10t\right)+cos\left(2\pi20t\right)+cos\left(2\pi50t\right) I use the online calculator to find me the FT of the signal, which is...
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    Subspace and Its 3 Conditions

    Homework Statement R = { (a+1, b 0) | a, b are real numbers} S = { (a+b, b, c) | a, b, c are real numbers) T = R intersect S I have shown that R and S are subspaces of R^3. Now I have to determine whether T is also a subspace of R^3. The answer provided is that yes, T is also a...
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    Linear Algebra - Subspace

    Homework Statement Show that S = {(a+1,b,0)|a,b are real numbers} is NOT a subspace of R^3. Homework Equations The Attempt at a Solution I take a specific counter example: Let k = 0 inside real, and u = (1+1,1,0) inside S ku = 0(1+1,1,0) = (0,0,0) not inside S So, S is...
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    Basis of Image of Linear Transformation

    Homework Statement Given a linear transformation F: R^3 --> R, F(x,y,z) = 3x-2y+z, find Im (F) and dim (Im (F)) Homework Equations I have found that dim(ker F) = 2 and from the theorem dim (V) = dim (Ker F) + dim (Im F), I know dim (V) = 3, so dim (Im F) = 1. The Attempt at a...
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    Dirichlet Problem on a Disc - Transfomation

    Homework Statement This is a problem I found in my note. It says that from u xx(subscript) + u yy = 0, we can arrive at u rr + (1/r) u r + (1/r^2) u theta theta = 0 by using the transformation x = r cos theta, y = r sin theta, r = square root of (x^2+y^2), theta = arctan (y/x). The...
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    Finding 1/n + 1/(n+1) + + 1/2n in terms of the Euler's constant

    Homework Statement By using the Euler's constant (0.577...), express the limit of the sum 1/n + 1/(n+1) + ... + 1/2n explicitly as n goes to infinity. Homework Equations The limit as n goes to infinity for [1/1 + 1/2 + ... + 1/n - In (n)] is the Euler constant, 0.577... It has been...
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    Finding Equations of Vector Line Perpendicular to another Vector Line

    Homework Statement Line 1's equation is given as r(t) = <t, t, 1-t>. Line 2 passes through the origin and is perpendicular to line 1. Find the equation of line 2. Homework Equations The Attempt at a Solution If it is a 2-D case then it would be a lot easier, since m1 x m2 = -1...
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    Mass Spectrometry

    Homework Statement An organic compound contains some C and H atoms and 1 S atom. Element sulfur, S has 2 isotopes, S-32 ans S-34. Their relative isotopic abundance is 100:4 (or 25:1). On the mass spectrometry graph, how would the peak of M+2 compared to that of M+'s peak? Homework...
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    Finding critical temperature from equation of state

    Homework Statement Given the equation of state PV=nRT. So, I get the first derivative and second derivative (P with respect to V) and equate them to 0. Then, I found out that they cannot be equal to 0, so I make the conclusion that no critical temperatue exists for an ideal gas. If the van...
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    Irrationality homework problem help

    Homework Statement Prove that for any n, sqrt n + sqrt (n+1) is irrational. Homework Equations The Attempt at a Solution Well, I know that sqrt 2 + sqrt 3 is irrational. How about sqrt n + sqrt (n+1)? Let sqrt n + sqrt (n+1) be rational. [sqrt n + sqrt (n+1)]^2 = (m^2/n^2)...
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    H3PO4 Stepwise Neutralization

    For the triprotic acid, phosphoric acid, H3PO4, there are Ka1, Ka2, and Ka3. And we know from the book that the value of Ka3 is too small, so we neglect the [H+] contribution from it. Normally, we encounter questions involving pH calculations of neutralization of a diprotic acid and a...
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    Algebra- Vector ce and subspace

    Algebra- Vector space and subspace Homework Statement Here are some true or false statements given in my test. (a) R^2 is a subspace of R^3. (b) If {v1, v2, ..., vn} is a set of linearly dependent vectors, then it contains a zero vector. (c) If {v1, v2, ..., vn} is a spanning set, then...
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    Chemistry Stereochemistry, molecules, internal symmetry

    Homework Statement Does the molecule in the file attached have an enantiomer? Homework Equations The Attempt at a Solution I think I can draw an internal symmetry plane to divide the molecule into 2 equivalent parts, but then again, it doesn't seem quite right to me. Building...
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    Comp Sci C++: Leap Year

    Homework Statement I just started my C++ courses and here is my question. Homework Equations The Attempt at a Solution //This program states whether a given year is a leap year or a non-leap year. #include <iostream.h> #include <conio.h> void main () { int year...
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    Hydroxide ion

    Question: State the hybridization used and the molecular geometry shape of hydroxide ion, OH-. Okay, so I draw out the Lewis structure, where there are 6 electrons (3 lone pairs) around the O atom, which is bonded to the H atom. So, I think base on the 3 nonbonding domains and 1 bonding...
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    Confusion of Concavity

    Given f'(x) = 1/(3+7x). Find the interval where the graph of f(x) concaves downward. First, I differentiate f'(x)= 1/(3+7x) to get f''(x) = -7/(3+7x)^2. So, for the graph to be concave downward, I put f''(x) = -7/(3+7x)^2<0, so 7/(3+7x)^2>0 (3+7x)^2 is always more or equal 0. Then, I...
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    De Broglie or Schrodinger?

    Here is an objective question: The first full presentation of the behavior of electron in hydrogen atom by using wave mechanics is by: Louis de Broglie or Schrodinger? As far as I know, Schrodinger is famous for his wave equation. Meanwhile, de Broglie is known for the equation lambda= h/mv.
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    Roots of Equation

    Question: How many real roots does the equation x^5 + x + c = 0 have on the interval of [-1,1]? I try to differentiate the equation, then I obtain 5x^4 + 1. When at its local minimum or local maximum, 5x^4 + 1 = 0. So, there is no solution for the equation, since 5x^4 + 1 > or = 1...
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    Special Electronic Configurations

    I ony know how to explain the anomality of Cr and Cu's electronic configurations. And in the assignments, I am asked the following: (a) For Rh: Why not [Kr]4d7 5s2 but [Kr] 4d8 5s1 (b) For Ir: Why not [Xe]4f14 5d8 6s1 but [Xe] 4f14 5d7 6s2 Can someone give a brief explanation...
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    Ionization energy and speed of electron

    A neon sign is a gas discharge tube in which electrons travelling from the cathode to anode collide with neon atoms in dicharge tube and knocks electrons off of them. As electrons return to the neon ions and drop to lower energy levels, light is given off. How fast would an electron have to be...
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    Circle in a parabola

    A circle with the radius of 1 unit is inscribed in a parabola with the equation of y=x^2. Find the coordinate of the center of the circle. The answer given in the book is (0, 5/4). I attempt this question by letting the equation of the circle be x^2 + (y-c)^2 = 1, where c is the...
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