0
votes
1answer
33 views

being $\mathbf{w}$ a vector, how do I calculate the derivative of $\mathbf{w}^T\mathbf{w}$?

Let's say that I have a vector $\mathbf{w}$. How can I calculate the derivative in the following expression? $\frac{\mathrm{d}}{\mathrm{d}\mathbf{w}}\mathbf{w}^T\mathbf{w}$
0
votes
0answers
25 views

being $\mathbf{a}$ and $\mathbf{b}$ two vectors with same length, how do I expand $(\mathbf{a}^T\mathbf{b})^2$?

Let's say that I have two vectors $\mathbf{a}$ and $\mathbf{b}$. Assuming that they have same length, their product $\mathbf{a}^T\mathbf{b}$ and its square $(\mathbf{a}^T\mathbf{b})^2$ are scalars. ...
1
vote
1answer
38 views

How do you prove a hilbert transform?

I am stuck with this question below, I need help;
0
votes
2answers
26 views

Find solutions for an differential equation system

I have a differential equation system $x_1'(t) = -x_2(t)$ $x_2'(t) = -x_1(t)$ I see that I can write $\dot{x} = Ax$ where $A = \begin{pmatrix}0 & -1 \\ -1 & 0\end{pmatrix}$ The complete ...
1
vote
1answer
38 views

The level set of a smooth function

Let $f$ be a smooth function on a manifold $M$. Fix a point $p\in M$ and let $df\in T^\ast_pM$ be the differential of $f$ at $p$. I read that the subspace of $T_pM$ of vectors $X$ such that $df(X)=0$ ...
0
votes
2answers
45 views

proving this algebraic expression

I want to prove that: $A^n-B^n=(A-B)(A^{n-1}+A^{n-2}B...+AB^{n-2}+B^{n-1})$ I checked that it holds for n=2, and n =3(not n=1). So I think maybe I can use induction? However I get stuck: ...
2
votes
2answers
58 views

Find the $L^2[-\pi,\pi]$ projection of $f(x)$

I need to find the $L^2[-\pi,\pi]$ projection of $f(x)=x^2$ onto the space $V_n\subset L^2[-\pi,\pi]$ spanned by ...
0
votes
1answer
28 views

Why is alternative sign in Hessian subdeterminant a necessary and sufficient condition for multivariable maxmization

The necessary and sufficient condition for a maximal point in a multivariable function is the following $$\text{i. } x \text{ must satisfy first order condition}$$ $$\text{ii. } |H|_1 < 0 \text{ ...
0
votes
2answers
58 views

Parallel vectors in $\mathbb{R}^n$.

Def: We say that $\vec{x},\vec{y}\in\mathbb{R}^n$ are parallel vectors if $|\vec{x}\cdot \vec{y}|=||\vec{x}||\,| |\vec{y}||$. (i.e equality holds in Cauchy–Schwarz inequality) I'm having some ...
7
votes
1answer
122 views

Quaternion integration

If the angular velocity is changing continuously, the following holds true $ q(t)=q(0)\exp\left({\int_{0}^{t}\frac{q_\omega(\tau)}{2}\ d\tau}\right) \tag 1$ Specifications and Data $q(t),q(0)$ ...
1
vote
1answer
46 views

Finding minimum point of a function using linear algebra

Given a function $$q(x,y)=2x^2-2xy +2y^2$$. Find the minimum point of the following function by first converting it to a matrix form and using the diagonalisation of the matrix to find its minimum ...
2
votes
1answer
30 views

Surjective function from a set of funtions to itself

Let a function be defined by $f \longmapsto f'$ acting from the set of all polynomials to itself. I am asked if this is surjective. I would like to think it isn't, but I'm in doubt how I should ...
1
vote
0answers
36 views

Preparation for a Linear Algebra Class

I have just entered my Junior Year as a CS student. While I have already taken discrete math and Theory of Computation, and have not found myself needing any additional math skills thus far; I ...
3
votes
1answer
53 views

Sums of special vectors

Let $v$ be a vector obtained by taking a sum of $k$ vectors the of the form $(0,0,\ldots,0, -n, *,*,\ldots,*)$, where $"*"$ stands for either $0$ or $1$, and the position of the $-n$ entry can vary ...
0
votes
0answers
40 views

Function with constant derivative

We have a column matrix $P_i$ defined as follows $P_i= {\begin{pmatrix} a_i \\ b_i \\ c_i \end{pmatrix}}_{3\times 1}\tag 1 $. Given Data All $a_i,b_i,c_i$ are constants It is given that $i$ can ...
2
votes
1answer
44 views

Sum involving integer part and cosine function

How to find the close form of sum and eliminate $k$? $$ \sum_{k=1}^{n} \frac{n \left[ \cos \left( \frac{n}{k}- \left[\frac{n}{k} \right]\right) \right]}{k} $$
3
votes
1answer
35 views

Integration by parts for Matrices

I understand how to do integration by parts for individual functions. I am trying to apply integration by parts to matrices/vectors where the order of terms is important. So say I have a matrix A ...
1
vote
1answer
59 views

Is it true that $d\textbf{S} = dy dz\textbf{ i }+ dx dz\textbf{ j }+ dx dy\textbf{ k }$

I came up with this in my mind, Just wondering if it is true I am thinking about it too, will post my observations, if any
3
votes
2answers
31 views

problem with denominator in transformation

hi i cant understand where the 2 comes from in this transformation any help would be appreciateD
0
votes
1answer
40 views

calculus / algebra

Hi can anyone go through the transformation of the equation below as i cannot understand where the 2 in comes from any help would be much appreciated $$\frac{\omega k^{0.5}}{\omega k} = ...
0
votes
1answer
62 views

Quaternion expansion

I have a quaternion equation $ \psi(s)=Pe^{\frac{1}{2}k(s)}\tag 1$ Given conditions and data Here P is a constant unit Quaternion defined for 3D rotation matrix as $(p_1,p_2,p_3,p_4) , p_4\in ...
4
votes
3answers
85 views

Intuitive and convincing argument that functions are vectors

Back to school time again. As I'm discussing all the mathy stuff and insights gained over the summer, I cannot help but notice that many of my peers in second or third year undergrad cannot bridge the ...
2
votes
0answers
67 views

Definite Integral involving matrices

We have a definite integral of the form given below $ f(t) = \int_0^1 e^{\alpha X(t)} \frac{dX(t)}{dt} e^{(1-\alpha) X(t)}\,d\alpha \tag 1$ Given Data in the question $X(t)$ is a ...
0
votes
1answer
30 views

Matrix Solution

I have matrix integral equation of the following form ${f^{'}(x)}_{1 \times 1}A_{3\times 3}=P_{3\times3} (1-x)+Q_{3 \times 3}x \tag 1$ . All dimensions are indicated in equation itself. " ' " ...
1
vote
0answers
42 views

Can integral transforms be viewed as change of basis formulas?

Forgive any lack of rigor, this question is kinda all over the place. If you have a set $B $ of $ N $ basis functions $ g_0(t), g_1(t), g_2(t), \dots, g_{N-1}(t) $ which are orthogonal over $[t_1, ...
0
votes
2answers
48 views

Matrix Exponent - equivalent of a rotation matrix

For every Rotation Matrix,there is a Matrix Exponent representation where the power is a skew symmetric matrix. More clearly if I have a rotation matrix ${R}_{3 \times 3}$ then there will be a skew ...
1
vote
0answers
64 views

Matrix exponent form

We have an equation of matrix exponent $ Ae^{Ax}R-e^{Ax}R (P_1 +P_2 x) = Y \tag1$ Given condition $A,R,P_1,P_2,Y$ are constant $3 \times 3 $ matrices. R is invertible,orthonormal,determinent ...
1
vote
1answer
44 views

Approximate Equivalent To Michael Spivak's text, “Calculus” but for Linear Algebra?

Does anyone know of an approximate equivalent To Michael Spivak's text, "Calculus" but for Linear Algebra? I love the way this book is written! It is simultaneously rigorous and thorough without ...
0
votes
0answers
44 views

Strictly Convex Functions

I am trying to show the equivalence of two definitions of strictly convex functions. Let $f:\mathbb{R}^n\to \mathbb{R}$ be a smooth function. The function $f$ is strictly convex if for each ...
1
vote
0answers
64 views

what does the second derivative of a linear function mean?

So if I have a function f(x) = 7x-2 the first derivative is 7 which I'm inclined to think that the second derivative ...
1
vote
1answer
169 views

Is it possible for a triangular matrix in echelon form to not have a unique solution and how?

I want to know if it is possible for a triangular matrix in echelon form to not have a unique solution and how? Isn't there something to do with the determinant that shows this? or am I wrong?
1
vote
1answer
67 views

An upper bound on a sequence of positive numbers $x_n$ such that $x_{n+1} \le \min \{b \cdot x_n,c\}$

Suppose $\{x_1, x_2,\ldots, x_n,\ldots \}$ is a sequence that satisfies $x_0 = a$, and $x_{n+1} \le \min \{b \cdot x_n,c\}$, where $a,b,c>0$ are constant given numbers, and $x_i>0$ for ...
1
vote
1answer
92 views

find a matrix transform

Given a vector $v={(v_1,v_2,...,v_n)}^T$, I would like to find some matrix operations on $v$ to create an $n \times n$ matrix $X$ such that its entry $X_{i,j} $ satisfy (1), (2), (3), (4), ...
-4
votes
2answers
53 views

Orthogonalization of two Vectors [closed]

Given two vectors $v_1$ and $v_2$, which have a given angle $\theta$≠ $$\frac {π}{2}$$, in between; How would one apply a Gram Matrix to define an inner-product, in order to orthogonalize the two ...
0
votes
0answers
42 views

ODE with multiple simple conditions $f'(x)=f(x)(Ax+D ) $

I have an ODE to solve . The main issue is,in addition to solving it I have to keep some conditions too in the solution of f(x).. I am bit confused regarding how to deal with it. Equation is given ...
0
votes
2answers
48 views

Solve nonliner equations

We are trying to find intersection of hyperbolas and we ended up in five equations $$\begin{align} A_1X^2+B_1Y^2+C_1XY+D_1X+E_1Y+F1&=0\\ A_2X^2+B_2Y^2+C_2XY+D_2X+E_2Y+F2&=0\\ ...
1
vote
1answer
54 views

Properties of the positive definite Hessian matrix of a convex function

I'm reading about nonlinear programming and I'm having trouble understanding the cool properties that a positive definite Hessian matrix $Q$ of $n$-dimensional function $f: \mathbf{R}^n\rightarrow ...
0
votes
4answers
37 views

Surjective function - proving

$f: \mathbb{R}\to \mathbb{R}$ $f(x) = x^3 -2x^4$ In order to prove that $f$ is not surjective, my teacher told me to find that in most the $f$ is negative. And indeed, only for $0<x<0.5$ it's ...
1
vote
1answer
33 views

Surjective functions and cal'

$f,g: \mathbb{R}\to \mathbb{R}$ Both are also surjective functions. My question is if $f+g$ will be also surjective. I need to dis/prove it if it's true or false. Now, my friend told me it's false ...
0
votes
0answers
45 views

How prove $S_{k}(x)=\sum_{i=1}^{n}x^k_{i}$ this System of equations The only solution?

when I read a china book,I see this follow interesting problem (the author says it is clear have follow) if give the number $S_{k}(x),k=1,2,3,\cdots,n$ ,and such $$\begin{cases} ...
1
vote
0answers
29 views

Conjunctive Normal Form representation/ First Order Logic.

in my research problem, I need to represent three types of three types of relationships between the variables x,y as the following:: " y Cooperates with x" relationship: means if there is two ...
1
vote
1answer
72 views

How to solve this graphing question?

$ \frac{|x-2|} {(x^2-4)}+\frac{(x-2)} {|x-2|} = b $ determine for which values of $b$ the equation has one and only solution. I tried sketching the graph, but was unable to do so accuratly...also, ...
-2
votes
1answer
71 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
3
votes
2answers
47 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
49 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
27 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
26 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?
2
votes
1answer
88 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
81 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
52 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 ...