# Tagged Questions

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? $\frac{\mathrm{d}}{\mathrm{d}\mathbf{w}}\mathbf{w}^T\mathbf{w}$
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. ...
1answer
38 views

### How do you prove a hilbert transform?

I am stuck with this question below, I need help;
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 ...
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 $df(X)=0$ ...
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: ...
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 ...
1answer
28 views

1answer
62 views

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 ...
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 ...
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 ...
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 ...
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 ...
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?
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 ...
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), ...
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 ...
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 ...
2answers
48 views

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 ...
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, ...
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
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 ...
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 ...
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 ...
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?
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 ...
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: $$z_1^6 +z_2^6+z_3^6+z_4^6+z_5^6+z_6^6$$ ...
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 ...