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...
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...
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...
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}...
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...
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 =...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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)...
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...
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...
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...
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...
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...
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...
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.
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...
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...
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...
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...