-2
votes
1answer
59 views

Integral question challenge [duplicate]

I try to find a reasonable solution for this equation but i couldent I try to study lots of material but i couldent solve it. I am a high school student and try to learn. Integral cos(log x)dx
2
votes
2answers
40 views

About matrix derivative

Suppose $A$ is a matrix with order n*n. we have the following equity but I don't know why. $f(x)=\frac{1}{2}x^TAx-b^Tx$. then $f'(x)=\frac{1}{2}A^Tx+\frac{1}{2}Ax-b$ Is there any rule like scalar ...
0
votes
1answer
46 views

Show that a linear mapping is invertible over all $\Bbb R^{2}$

Show that (under appropriate assumptions) a general linear mapping $F(x,y) = (ax+by,cx+dy)$ is invertible over all of $\Bbb R^2$ (i.e. there is a single inverse for all of $\Bbb R^2$). What ...
0
votes
1answer
25 views

Properties of $f(x) = \det (A+xB)$

Let $A_{n \times n},B_{n \times n}$ be real square matrices. Let $f(x) = \det (A+xB)$. Then if n is odd, then $f(x)$ has inflection point $f(x)$ doesn't have a horizontal asymptote ...
0
votes
0answers
13 views

Solution of definite integral of product of bessel function and exponential

I have an integral $I=\int_{\theta} \int_r J_m(k_1r)e^{-j[P_x r \cos(\theta)+P_y r \sin(\theta)]} r dr d\theta$ $0\leq\theta\leq2\pi; r<\infty$ is there any method to solve this?
1
vote
1answer
68 views

First Order Logic Consistency Big Problem

as i read some tutorial material on First Order Logic, i deduce that the following formula was consistent in FOL except the third one. am i right? i have doubt about the first one. any idea? thanks to ...
0
votes
3answers
78 views

Complex Roots and calculations

roots of the equation $z^6 =1-\sqrt3 i $ are $$z_1,z_2,z_3,z_4,z_5,z_6 $$ calculate:$$|z_1|^3 +|z_2|^3+|z_3|^3+|z_4|^3+|z_5|^3+|z_6|^3$$ also calculate: $$z_1^6 +z_2^6+z_3^6+z_4^6+z_5^6+z_6^6$$ ...
0
votes
1answer
47 views

How to simplify linear algebra equation

Im a trying to understand the derivation of an linear algebra equation. It is from a paper about 3D mbICP scanmatching. I am not that good at linear algebra but I am trying to learn. The equation ...
0
votes
0answers
31 views

inverse of function with sine mooculus

I'm trying to do a calculus course on line: mooculus and I'm trying to answer this question: The height in meters of a person off the ground as they ride a Ferris Wheel can be modeled by h(t) = ...
0
votes
0answers
15 views

How to find iso function value points without exploring all points in 2D space

Consider a 2D graph with dim1 and dim2 represented as X and Y respectively. The range of X and Y are 1 to 100. Hence there are 10000 points in the 2D space. Each point in the space is some function of ...
1
vote
1answer
37 views

Computing the kernel and image of a linear transformation

I have a linear transformation $f:R^3 \to R^3$ , $f(x,y,z) = (2x+2z, x+z, x + 3y -2z)$. I need to find out Ker and Im of $f$. I found out $\ker(f) = \{(-a,a,a) \mid a \in \mathbb R\}$; ...
0
votes
1answer
25 views

how to find iso-cost contours on a 2d plot efficiently

Consider a 2D plot in which dimension 1 and 2 represent quantity 1 and 2 respectively ranging over 0 to 100. Each point in the space corresponding to (x,y) represent cost of choosing quantity 1 as x ...
-1
votes
0answers
17 views

Linear Equation Formulas for specific questions

Im trying to figure out how to do this problem, but it is just extremely confusing to understand how too do. A cricket chirps at different retes depending on temperature. You can estimate the ...
1
vote
1answer
91 views

Vector function tough question

If a curve has the property that the position vector $\vec{r}(t)$ is always perpendicular to the tangent vector $\vec{r'}(t)$, how can I show that the curve lies on a sphere with center the origin? ...
6
votes
2answers
465 views

Why integration operator has no eigen values?

Let $V$ be the vector space of all functions from $\mathbb R$ into $\mathbb R$ which are continuous. Let $T$ be the linear operator on $V$ defined by $$(Tf)(x) = \int_0^x f(t) dt$$ Prove that ...
8
votes
3answers
81 views

Find the expansion for $\det(I+\epsilon A)$ where $\epsilon$ is small without using eigenvalue.

I'm taking a linear algebra course and the professor included the problem that prove $$ \rm{det}(I+\epsilon A) = 1 + \epsilon\,\rm{tr}\,A + o(\epsilon) $$ Since the professor hasn't covered the ...
2
votes
2answers
182 views

Prove that the Laplace trasform is a Linear trasformation

Could you help me prove that the Laplace Trasform is a Linear trasformation? Thank you.
0
votes
0answers
19 views

is this a valid counter-example - function is not locally invertible

Let $S_n$ be the set of all symmetric matrices with real entries of size $n$x$n$. We are asked if the function $f:S_n \to S_n$, $f(A)=A^2$ is locally invertible for every $A$ (Using the Inverse ...
3
votes
0answers
17 views

$\sqrt{X}$ where $X$ is a positive definite matrix is smooth $C^{\infty}$ [duplicate]

I'm trying to prove the following statement. Let $P_n \subset Mat_{nxn}(\mathbb R)$ be the set of all symmetric positive definite matrices with real entries of size $n$x$n$. Let $\sqrt{}:P_n \to ...
0
votes
0answers
27 views

Subset Sum represented as a perfect number

Can we form a set of $29$ distinct integer elements such that every subset of elements possible has a sum which is a perfect power? A perfect power is a positive integer that can be represented a p^q ...
0
votes
1answer
36 views

Why is this proof valid - inverse function theorem

Question from worksheet, I don't fully understand the solution the teacher gave. Question: let $S$ be the set of symmetric positive definite matrices of dimension $n$x$n$. Let $T: S \to S$, ...
2
votes
0answers
18 views

Rank of the differential

Let $f:\mathbb R^n \to \mathbb R^n$ such that $f$ maps roots of a polynomial to its coefficients. Meaning: if $(x-x_1)(x-x_2)...(x-x_n)=x^n+a_1x^{n-1}+a_2x^{n-2}+...+a_n$ then $f\begin{pmatrix} x_1 ...
0
votes
0answers
49 views

Finding criteria for a household financial budget falsification

I’m working on a financial problem about budget of households. Households in a state fill a form about their net budget in every year and our insurance company investigate their financial status and ...
0
votes
1answer
24 views

Derivative of a Matrix with respect to a vector

I know that for two k-vectors, say $A$ and $B$, $\partial A/\partial B$ would be a square $k \times k$ matrix whose $(i,j)$-th element would be $\partial A_i/\partial B_j$. But could someone please ...
0
votes
2answers
65 views

Basis of solution sets

I know that the collection of all solutions to $\sum_{i=0}^nA_iy^{(i)}(t)=0$ form a vector space. But in which way can one find out its basis? Of course I already learned what the basis is. But the ...
0
votes
0answers
23 views

How do I express this profit function as a function of prices only?

How do I express this profit function as a function of prices only? The function is $(p-AC)q$, where $p$=price, $AC$=average cost (NOT marginal cost), and $q$=quantity sold. so in numbers ...
0
votes
1answer
71 views

Weighted Singular Value Decomposition

Lemma: $\forall A\in R^{n\times n}$ and a diagonal matrix $\forall W\in R^{n\times n}$ with $ w_{11}\geq w_{22}\geq ...\geq w_{_{nn}} >0$. The singular value decomposition of A denoted by: $A=XM ...
1
vote
2answers
60 views

Integral with matrices?

I have $\int_{-\infty}^\infty \int_{-\infty}^\infty exp(-x^T Ax) \mathrm{d}x_1 \mathrm{d}x_2$ $A = \left[ \begin{align} 3 && 2 \\ 2 && 3 \end{align} \right]$ Where $x^T = (x_1,x_2)$ ...
0
votes
0answers
26 views

How can I find the coefficients in a orthonormal linear combination with the most accuracy?

So I have a set of functions in spherical coordinates $f_k(r,\theta,\phi)$ and $g(r,\theta,\phi)$. Both functions sets are real and defined in the unit ball and I want to write the function ...
0
votes
1answer
43 views

Finding the integral of a trig function using a matrix

It can be shown that Ɓ = {1, $\cos(t)$,…$\cos(6t)$ and Ƈ = (1,$\cos(t)$,…$\cos^6$(t)} span the same subspace of Ƈ(ℝ) a. Use an appropriate change of coordinate matrix to find $cos^6$(t) in terms of ...
0
votes
1answer
12 views

Showing that the set $\displaystyle \{e^{2\pi i (mx+ny)}\}_{n,m=-\infty}^\infty$ is an orthonormal set

I have the following problem: Show that the set $\displaystyle \{e^{2\pi i (mx+ny)}\}_{n,m=-\infty}^\infty$ is an orthonormal set in $L^2(D)$, where $D$ is any square whose sides have length ...
1
vote
2answers
50 views

Why is there an “absolute value” and a norm in the Schwarz Inequality?

This really bothers me, and I'm not sure if it's just that I'm not understanding it correctly. For the moment, assume we are working in a vector space $V$ over $\mathbb{R}^n$. Let $x,y \in V$. We have ...
1
vote
3answers
40 views

Understanding dot product of continuous functions

I'm reading about Fourier analysis and in my book the author speaks about dot product for continuous functions $f, g\in L^2(a,b)$(the set of functions which are square-integrable on the interval ...
1
vote
2answers
29 views

slope of a line in 3D coordinate system

Suppose I have $2$ points in a 3D coordinate space. Say $p_1=(5,5,5)$, $p_2=(1,2,3)$. How do I find the slope of the line joining $p_1$ and $p_2$? After getting the slope (which I assume will be an ...
0
votes
1answer
34 views

Derivative of $f(x)=\|Ax\|_2^2$

I'm trying to find the derivative of $f(x)=\|Ax\|_2^2$ where $A$ is some matrix and $\|u\|_2$ is the euclidean norm of $u$, $\|u\|_2 = \sqrt{u_1^2+u_2^2+\cdots+u_n^2}$ I know how to do this by ...
1
vote
1answer
24 views

Proving that orthonormal set is an orthonormal basis

If I know that the set of functions $\{\phi_n\}_1^\infty$ forms an orthonormal basis on $L^2(a,b)$ and the set $\{\psi_n\}_1^\infty$ is an orthonormal set on $L^2(\frac{a-d}{c}, \frac{b-d}{c})$, with ...
1
vote
1answer
21 views

Proving that a set $\{\psi_n(x)\}_1^\infty = \{\sqrt{c}\;\phi_n(cx+d)\}_1^\infty$ is an orthonormal basis

I have the following problem I need to solve: Suppose $\{\phi_n\}_1^\infty$ is an orthonormal basis in $L^2(a,b)$ (set of square-integrable functions on $[a,b]$). Suppose $c>0$ and $d\in ...
0
votes
1answer
34 views

Gram-Schmidt method and matrices help please!

How would I use the method of Gram-Schmidt to obtain an orthonormal basis for the column space of the matrix? Any help is appreciated!
2
votes
1answer
252 views

How to make this polynomial the zero polynomial?(recursively)?

Given a fixed $\beta \in \mathbb{R}$, I want to find the $c_0,...,c_n$ for arbitrary $n \in \mathbb{N}$ such that the polynomial \begin{align}P_n(z):=z(1-z) ...
1
vote
0answers
23 views

Jacobian in Change of Variables

Let us consider an integral $\int \mathrm{d} ^ 4 k _ {2} \int \mathrm{d} ^ 4 k _ {1} \, f (k _ {1}, k _ {2})$, where $k _ {1}$ and $k _ {2}$ are four-dimensional vectors in Euclidean space. We want to ...
4
votes
1answer
131 views

Product of Elements in SU(2)

Let $$ V := \frac{x_4+i\vec{x}\cdot{\vec{\sigma}}}{\left|x\right|}$$ where $\left(x_1,x_2,x_3,x_4\right)\in\mathbb{R}^4$, $|x|$ is the Euclidean norm, and $\sigma^j$ are the Pauli matrices. Let ...
0
votes
0answers
24 views

5 Parameter Affine Transformation

I am working on computing affine transformation using Gradient Ascent Method, so the Inverse compositional algorithm. However, I am stuck in one probably simple step but I fail to understand them. ...
2
votes
2answers
36 views

Solving bernoulli differential equation

How to solve $$t \frac{dy}{dt} + y = t^4 y^3$$ First I divided by $t$ to get $$\frac{dy}{dt} + \frac{y}{t} = t^3 y^3$$ Then I multiplied through by $y^{-3}$ to get $$y^{-3} \frac{dy}{dt} + ...
1
vote
1answer
35 views

How to find Area and Perimeter

hey guys can you solve these 2 questions for me shown in the image below. I've done this but I think it's not correct could you guys just show me the solution with working. Thanks in Advan =) God ...
0
votes
1answer
44 views

Reduced row echelon form with variables

I'm new to this, but if I have the matrix \begin{equation} A= \begin{bmatrix}1&2&3&1\\2&1&1&x^2+x \\ 3&6&x&x-6\end{bmatrix}\end{equation} and if I want to use the ...
0
votes
2answers
63 views

Eigen values and Eigen vectors

Let A be a 4x4 matrix with real entries such that $ \ -1,1,2,-2 \ $ are its eigen values.If $B=A^4-5A^2+5I$ ,where $I$ denotes the 4x4 identity matrix ,then which of the following statements are ...
1
vote
1answer
25 views

Prove uniqueness of polar-coordinates $(R>0, \theta)$ up to angle $\theta+ 2 \pi$.

Prove uniqueness of polar-coordinates $(R>0, \theta)$ up to angle $\theta+ 2 \pi$. Suppose we have $(x,y) \in \mathbb R^2$. Then we can transform this point to polar-cordinates $(R>0, ...
1
vote
2answers
79 views

Finding a particular solution to the non-homogenous system

I have the following problem $\vec{x}^{'}(t)=\begin{pmatrix} 2 & -5\\1 & -2 \end{pmatrix}\vec{x} + \begin{pmatrix} \csc t\\ \sec t \end{pmatrix}$ Step 1) Find the Eigenvalues ...
0
votes
1answer
59 views

List of topics for basic calculus (1st,2nd,3rd semester)

I am an computer science student, currently studying in 2nd semester. Therefore my math courses are pretty weak. Although I "aced" them, I still feel I could use some extra basic calculus knowledge in ...
0
votes
1answer
23 views

Is $hh^T$ positive semi-definite ($h$ is a column non-negative vector)? [duplicate]

Is $hh^T$ positive semi-definite? It seems to be positive semi-definite, but I cannot prove it. Please help:)