0
votes
1answer
25 views

Metric for vector sets

I am currently working on a classification algorithm. Each class is represented by a set of 3D vectors. The cardinality differs for each class. The order of the vectors in a set is completly random. ...
0
votes
1answer
25 views

Matrices as sets of vectors

What exactly does it mean when someone says a matrix may be intrepreted as a set of vectors? As in: "A matrix can be considered a set of vectors, organised as rows or columns" It seems it would only ...
0
votes
1answer
33 views

Finding the area of a triangle from vertices? Linear Algebra

I pretty much did this problem, but I failed to get the few last blanks where they ask the area. Its confusing, they say its half the volume of matrix (u v w) in the start of the question. which means ...
0
votes
4answers
46 views

algebraic representation of a line in 3d

Is an algebraic representation of a line in 3d possible, or there can be only a parametric one?
2
votes
1answer
46 views

Vector calculation question

the points a b c d are concordantly ( 1,2,-3) , (-1,2,1) , ( 0,1,-2) , ( 2,-1,1) find formula of the plane going thorugh d and which is pararlel to plane abc calculate the volume of pyramid abcd. ...
1
vote
1answer
103 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? ...
0
votes
1answer
49 views

Finding loci of possible points satisfying vector simultaneous equations

I recently had an exam and a question came up which I was only partially able to answer. The question was the following: Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be constant vectors in ...
1
vote
0answers
31 views

How to estimate/determine surface normals and tangent planes at points of a depth image (point cloud)?

I have depth image, that I've generated using 3D CAD data. This depth image can also be taken from a depth imaging sensor such as Kinect or a stereo camera. So basically it is depth map of points ...
0
votes
3answers
59 views

Find eigenvector of the linear operator

Task is to find an eigenvector of the following linear operator: $f \to \int^{x}_{-x} f(t)dt$ in the linear span $\langle cos(x), sin(x), ...,cos(mx),sin(mx)\rangle$. I know how to find eigenvectors ...
2
votes
0answers
32 views

Interior Products

Over on the Wiki page for interior products: http://en.wikipedia.org/wiki/Interior_product There is a line that says $\iota_X \alpha = \alpha(X) = \langle \alpha,X \rangle$ where $\alpha$ is a ...
2
votes
3answers
260 views

Notation: subscript vs. superscript for coordinate vector fields

Some books write the coordinate vector fields with a subscript as $$\frac{\partial}{\partial x_i}$$ while some write it with a superscript as $$\frac{\partial}{\partial x^i}.$$ Is there a conceptual ...
2
votes
1answer
26 views

Why are these two statements about vector products equivalent?

Let $w_1, \dots, w_m \in \mathbb{C}^d$. Condition (1) is: $\sum_i |\langle v, w_i \rangle |^2 = \eta$ whenever $\|v\| = 1$. Condition (2) is: $\sum_i u_i u_i^* = I^d$, where $u_i = w_i / ...
1
vote
1answer
61 views

$X_0\subset X$ finite codimension. Show there exists subspace $Z$ s.t. $X= Z\oplus X_0$

I'm not so familiar with these types of arguments concerning vector spaces, and direct sums. Let $X$ be a vector space and $X_0\subset X$ with finite codimension, i.e. $dim(X/X_0)$ is finite. Show ...
0
votes
1answer
54 views

Is this statement about vectors true?

If vectors $A$ and $B$ are parallel, then, $|A-B| = |A| - |B|$ Is the above statement true?
3
votes
1answer
4k views

Proof relation between Levi-Civita symbol and Kronecker deltas in Group Theory

In order to prove the following identity: $$\sum_{k}\epsilon_{ijk}\epsilon_{lmk}=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$$ Instead of checking this by brute force, Landau writes thr product of ...
3
votes
1answer
150 views

Different geometrical concepts of vectors

I'm a bit confused about the various geometric concepts of vectors. I'm mainly trying to understand if we can classify any vector into one of two categories.The first category would be free ...
0
votes
3answers
218 views

Geometrical Proof of a Rotation

I wanna prove geometrically ( and not by linear algebra, doing transformations in the bases ) the result of the rotation of a point. The proof should only include geometrical steps like using ...
0
votes
0answers
26 views

A vector map $f$ with the $f(k_1, .., ..k_n) + f(k_{n+1}, .., k_{2n}) = f(k_1+k_{n+1}, …, k_n + k_{2n})$

Is it possible to have a vector map $f: V^n \rightarrow \mathbb{R}$ with the $f(k_1, .., ..k_n) + f(k_{n+1}, .., k_{2n}) = f(k_1+k_{n+1}, ..., k_n + k_{2n})$? If so, is there a name given to this type ...
1
vote
1answer
75 views

Finding distances/similarity between vectors

Let's say for example I have these three vectors representing these three fields: $$\begin{array}{c|c|c} \text{Cars Sold} & \text{Cars in Lot} & \text{Profit Generated}\\ \hline \\140 & ...
3
votes
3answers
110 views

Is there a counterexample to the claim that if $\mathbf y\cdot\mathbf y=1$ and $\mathbf x\cdot\mathbf y=c$ then $\mathbf x=c\mathbf y$?

Let $x,y$ be arbitrary vectors where $\mathbf{y} \cdot \mathbf{y} = 1$ and $c$ be a real valued scalar. If $\mathbf{x} \cdot \mathbf{y} = c = c (\mathbf{y} \cdot \mathbf{y} ) = (c \mathbf{y} ) \cdot ...
3
votes
1answer
436 views

Cosine similarity between complex vectors

Having a pair of vectors where its elements are complex, is that possible to calculate the cosine similarity between these two vectors to see how similar they are?, and if the result of this ...
5
votes
3answers
166 views

Find the necessary and sufficient conditions on $A$ such that $\|T(\vec{x})\|=|\det A|\cdot\|\vec{x}\|$ for all $\vec{x}$.

Consider the mapping $T:\mathbb{R}^n\mapsto\mathbb{R}^n$ defined by $T(\vec{x})=A\vec{x}$ where $A$ is a $n\times n$ matrix. Find the necessary and sufficient conditions on $A$ such that ...
3
votes
1answer
91 views

Vector Equation spelled out

I am looking to find the $ x_1, y_1 $ coordinate that is distance $d$ from a starting point of $x_0, y_0$ coordinate as traveled along a line ($y=mx+b$) I was going to ask this question here, but I ...
-2
votes
1answer
441 views

Find if possible an orthogonal unit vector at: 2i + 3j - k and - 2i - 3j + 4k

The question is: Find, if possible, an orthogonal unit vector at: $2i + 3j - k$ and $-2i - 3j + 4k$. $$\left|\begin{matrix} i & j & k \\ 2 & 3 & -1 \\ -2 & -3 & 4 ...
0
votes
0answers
53 views

Knowing $\alpha$ and $\beta$, compute $\gamma$. $vers(\vec v)=(\cos\alpha,\cos\beta,\cos\gamma)$

Knowing that: $\vec v=|\vec v| vers(\vec v)$ $\cos\alpha=\frac{\vec v \cdot \vec i}{|\vec v|\cdot|\vec i|}$ $\cos\beta=\frac{\vec v \cdot \vec j}{|\vec v|\cdot|\vec j|}$ $\cos\gamma=\frac{\vec v ...
0
votes
2answers
208 views

Vector Analysis & Linear Algebra

I'm given a positive number, a unit vector $u \in \mathbb{R} ^n $ and a sequence of vectors $ \{ b_k \} _{ k \geq 1} $ such that $|b_k - ku| \leq d $ for every $ k=1,2,...$. This obviously implies $ ...
3
votes
2answers
11k views

Find the equation of a line which is perpendicular to a given vector and passing through a known point

There is given a vector $2 \vec i + \vec j - 3 \vec k$ and now I want to find the equation of a line that is perpendicular to the given vector and passing through a known point $(1,1,1)$. How can I ...
1
vote
1answer
95 views

Vector derivatives, what is the minimum of this matrix equation?

I am new to vector derivatives and trying to figure out a lot for my Machine Learning course. I have given the following: $x \in \mathbb{R}^n$, $y \in \mathbb{R}^d$, $A \in \mathbb{R}^{d \times n}$, ...
3
votes
2answers
4k views

mathematical difference between column vectors and row vectors

I'm writing a mathematical library; and I have an idea where I want to automatically turn column matrices and row matrices to vectors, with all of the mathematical properties of a vector. Answer I'm ...
-1
votes
1answer
238 views

How to extract d from a plane being created by two vectors?

I want to let fall a perpendicular from a point A in space being given by $A_x, A_y$ and $A_z$ on a plane being given by two vectors $B$ and $C$. Ultimately I want to determine the foot x0 of the ...
0
votes
0answers
75 views

clever solution to decomposition of linear products?

There may be a better name for this class of problem, and if so feel free to edit! Imagine a matrix consisting of the following columns: daily return, $\alpha_t$, $factor^1_t$, $factor^2_t$, ... and ...
0
votes
1answer
499 views

Are these three ways of normalizing a set/vector of numbers 'equivalent'?

Say I have a set of positive numbers (well, 0 is allowed, but if 0 is the max then no further operation is done and hence the special case of divide by zero won't arise). These are essentially some ...
2
votes
1answer
122 views

Example of a vector norm for which $\|I\|<1$

In order to prove a larger assumption, I need to find a vector norm over $M_n$ such that $\|I\| < 1$. None of the standard $p$-norms, nor the infinity norm work. I know that for matrix norms, ...
1
vote
1answer
64 views

Equivalence of a Vector Norm being Absolute

I'm trying to show that a vector norm $\|\cdot\|$ being absolute ($\|x\| = \|\;|x|\;\|)$ is equivalent to showing that $\|x'\| = \|[\alpha_1x_1\ldots\alpha_nx_n]^T\| = \|x\|$ for all ...
3
votes
3answers
1k views

Two 2d vector angle clockwise predicate

I've got three vectors v1[0,1], v2[-1,1], v3[1,1]. I calculated the angles beetween v1-v2 and v1-v3 using the forumla dot(vec1, vec2) / ||vec1|| * ||vec2|| and the both angles are the same (45 ...
2
votes
2answers
172 views

equation in matrix form

i have an equation $$\sum_{i=0}^m [w^{(i)}[(c-a^{(i)})-\frac{d^{(i)}(c-a^{(i)})\cdot d^{(i)}}{dis^2}]]=0$$ Where: $a,b$ are two end points in a 3D Line. $d$ is a vector and vector $d=b-a$. the ...
1
vote
2answers
41 views

How to define the new vector in my case?

(I am applying the things in a software, so I used some programming code, but they are easy to understand I think ) If I have a vector: ...