Questions tagged [geometric-interpretation]

Questions about understanding a problem geometrically.

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Any geometric interpretation for the adjoint system of a linear dynamical system?

On page 26, Section 1.3, of his book on linear dynamical systems1, Professor Roger Brockett asks: If $$\dot{\mathbf{x}}(t) = A(t) x(t) , \qquad \mathbf{x}(0) = \mathbf{x}_0$$ and $$\dot{\mathbf{p}}(t)...
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Geometric interpretation of $A^TA$

For a transformation $A \in \mathbb{R}^{n\times m}$ what exactly is the geometric interpretation of the transformation $A^TA$. If I understand it correctly the entries of $A^TA$ are the inner products ...
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Covariance Matrix outer product of vector itself

Let's define vectors as $x_1,x_2,...,x_m$ with zero mean where $R^d$ The computing covariance matrix is $\sum=\frac{1}{m}\sum_{i=1}^mx_ix_i^T \in R^{dxd}$. We know that the covariance matrix ...
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Geometric interpretation for norms and traces in number fields or matrix algebras

In a simple algebra $A$ over $\mathbb{Q}$ (one might just think of a number field or a matrix algebra), the trace can be interpreted geometrically as giving an "inner product" for $A$ as a ...
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Geometrical meaning of $x^2+y^2+z^2-xy-xz-yz$

I am looking for a geometrical interpretation of the symmetrical expression $$f=x^2+y^2+z^2-xy-xz-yz\tag{1}$$ with $x,y,z \in \mathbb{R}$. I could for example $f$ interprete as dot products of a ...
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Is there a geometric interpretation of $\int_a^b\left(x-\frac{a+b}{2}\right)f'(x)dx$

For a differentiable function $f:\mathbb{R}\rightarrow\mathbb{R}$, using integration by parts, we get $$\frac{(b-a)(f(a)+f(b))}{2}-\int_a^bf(x)dx=\int_a^b\left(x-\frac{a+b}{2}\right)f'(x)dx$$ The LHS ...
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What is the geometrical interpretation of a correlation matrix?

Assume we have $n$ unit norm vectors each of dimension $p$, where $p > n$. These vectors can be stacked together to form a matrix $M \in \mathbb{R}^{p\times n}$. Their Pearson Correlation matrix ...
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2 answers
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What does "interpret the point" mean?

As far as I understand, interpreting a function means finding its vertex, determining its shape and the direction, in which the function's hands are pointing. For example, if I am asked to interpret ...
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Understanding the Graph of a Multinomial Distribution

I am trying to understand exactly what information the graph of a multinomial distribution is supposed to convey. The thing I find strange is that a binomial distribution is graphed in two dimensions ...
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How to interpret strongly/weakly infinite dimensional spaces and what are some examples?

I have a question regarding strongly infinite dimensional spaces. Loosely speaking, $X$ is called strongly infinite dimensional if any pair of closed disjoint sets can be separated by a subset $L_i$ ...
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Geometrical interpretation of variance in terms of moments

Suppose $X$ and $Y$ are discrete random variables such that $$Y = aX+b$$ where $a$ and $b$ are scalars. It is easy to find a geometric interpretation of the following two facts $$\Bbb E[Y] = a \,\Bbb ...
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What is the geometric interpretation of $ \frac{1}{2}||a||^2\leq \langle a,b\rangle $?

Give $a, b$ in $\mathbb{R}^n$. What is the geometric interpretation of the following? $$ \frac{1}{2}\|a\|^2 \leq \langle a,b\rangle $$ In other words, what criteria should $a$ and $b$ have to satisfy ...
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What is the geometric interpretation of matrix addition?

I was studying linear algebra and trying to get a visual "feel" for it through watching 3Blue1Brown's "Essence Of Linear Algebra" series here Essence Of Linear Algebra Here, matrix ...
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Geometric interpretation of row operations on matrix (when solving systems of equations)

I understand that 2D matrix representing the lines in 2D space gives a unique solution where those lines intersect. Same in 3d, unique solution is where the planes intersect. Can someone explain what ...
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Generalizing the geometric interpretation of dot product to simple $k$-vectors

Background: For $u, v \in \mathbb R^n$, the dot product $u \cdot v$ can be interpreted geometrically as follows: Its magnitude is the product of the lengths of $u$ and $\operatorname{proj}_{u} v$. ...
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Philosophy of Integration (geometric interpretation): the difference between dx & a point?

This may sound like a stupid question, but if you're familiar with "Infinite Hotel Paradox", probably it won't be; So here we go: Integration of a scalar function $f: \mathbb{R} \rightarrow \...
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The radius of the circle which just touches the outer circle

Suppose two concentric circles with radius $a$ and $b\ (>a)$ and origin as their center. I wanted to put another circle whose center lies in a line $x=a$ (that is red line) in such a way that it ...
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Where does the 3rd power of the denominator of the curvature of the graph of a function come frome?

According to Wikipedia, the curvature of the graph of a function $f$ is given by the following ratio (assuming second-differentiability). $$\mathscr{k}_f(x) = \frac{f^{\prime\prime}(x)}{(1 + [f^{\...
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Remove outliers from a noisy curve

Allow me to present some images so that I may explain my problem. The images on the left contain a smooth curve surrounded by noise. Is there any approach that would help in eliminating the noise and ...
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Geometric interpretation of differentiability

I know that the geometric interpretation of differentiability for a function $f:\mathbb{R}^2\to \mathbb{R}$ in a point $(x_0,y_0)$ is that it admits a tangent plane in the point $P=(x_0,y_0,f(x_0,y_0))...
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Why doesn't a line passing through origin satisfy the 2 intercept form of the equation of a straight line?

The 2 intercept form of an equation of a straight line, is: x/a + y/b = 1 Where, a and b are the x and y intercepts respectively. For a line passing through origin, these two intercepts will become 0, ...
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1 answer
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Geometric interpretation of the non-uniqueness of the solution of a system of linear equations

Show that the equation $$\pmatrix{3&-7&0\\2&2&5\\1&3&4}\pmatrix{x\\y\\z} = \pmatrix{3\\2\\1}$$ does not have a unique solution, and give a geometrical interpretation. Now, the ...
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What is the Point, and Meaning, of the Mean Value of a Function?

Ok, so I know that the mean value of a function, $f(x)$, on the interval $[a,b]$ is given by (or defined by?) $$\frac{1}{b-a}\int_a^bf(x)~dx$$ but I have $2$ basic questions about this: $1$: From a ...
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5 votes
1 answer
725 views

Where is the "Interior" of a Clifford Torus

[I do not believe I have tagged this entirely correctly, feel free to change them.] In the case of spheres there is a related structure called a ball which based on its definition I visualise as being ...
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Is there a geometric interpretation for $\det(\nabla X)$?

Let $X$ be a vector field on a Riemannain manifold $M$. Consider $\nabla X:TM \to TM$, where $\nabla$ is the Levi-Civita connection of $M$. We know that $\operatorname{tr}(\nabla X)=\operatorname{div} ...
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Dirac Delta Function of a complex variable

Is there a context where delta(x+i*c), where c is a real number, makes sense? It came up while I was doing an Inverse Fourier Transform, and I failed to appreciate its significance. Does anyone know ...
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Geometric interpretation of linear programs with both inequality and equality constraints

I recently do a homework with linear programming, the standard form as: $$\begin{array}{ll} \text{minimize} &\displaystyle \mathbf{c}^T\mathbf{x}+\mathbf{d}\\ \text{subject to} & \mathbf{A}\...
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Geometric solution of $au_x+bu_y+cu=0$

This is a question that I read, and I am curious about how to solve $au_x+bu_y+cu=0$ by interpreting geometrically. Additionally, the second answer on that page says Let $v(x,y)=e^{cx}u(x,y)$ and ...
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Trying to Understand Time Interpolate

I am trying to understand pandas interpolation, time interpolate. I know that it is the same as linear interpolation if the time indexes are equally spaced. But I don't understand how it works when ...
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3 votes
1 answer
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Geometrical interpretation of matrix $A-B$

Is there a geometrical interpretation of subtraction of two matrices, with a special case of $I -A$ (subtraction of a matrix from identity matrix)? Reference: If $A$ is an idempotent matrix, the range ...
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2 votes
0 answers
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Is there a geometric interpretation of eigenvalues of integer matrices?

In some instances, like physics, you may find that quantities you are after are eigenvalues of matrices. However, for example, explaining that "the mass of a muon is an eigenvalue of a matrix&...
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Geometric intuition of the solution to $A \bf{x} = \bf{b}$

I am seeking the geometric intuition behind the following two cases using matrix $A_{m \times n}$ with rank $r$. When $m>n>r$, $A\textbf{x} = \textbf{b}$ has a solution if $\textbf{b}$ is in $C(...
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5 votes
0 answers
93 views

Geometric intepretation of transpose of matrix [duplicate]

Whenever we see a matrix $A=\bigl( \begin{smallmatrix} 3 & 2 \\ 1 & 2 \end{smallmatrix} \bigr)$ and $v=(3, 2),$ we can visualize that $(3, 2)$ represent the coordinates of $\mathbf i$ vector ...
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1 vote
0 answers
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basic level maths to geometrical interpolations (trilinear, prism, pyramid, tetrahedral) - where do I start?

I'm attempting to read Computational Color Technology by Henry R. Kang but I only have a GCSE level understanding of maths (if even that, it was 15 years ago..). I'd like to learn about geometrical ...
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1 vote
1 answer
157 views

A question about derivatives between Euclidean spaces: how should we construct it and interpret its definition?

As it is known from the single-variable calculus, given $X\subseteq\textbf{R}$, a function $f:X\to\textbf{R}$ and a adherent point $x_{0}\in X$ which is also a limit point, we define the derivative of ...
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How can 2 independent variables lead to a single variable parametrization? (Intuition)

Take the following formula: $$ \mu_{ij}=\left(\alpha_i\Sigma_i^{-1}+\alpha_j\Sigma_j^{-1}\right)^{-1} \left(\alpha_i\Sigma_i^{-1}\mu_i+\alpha_j\Sigma_j^{-1}\mu_j\right). $$ Where the alphas are ...
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What it is the interpretation of the below picture in set theory?

somone sent me the below picture and He asked me to give him its interpretation regarding set theory ? but I ask alos about its geometricall interpretation ?
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2 votes
1 answer
109 views

Examples about Cohen-Macaulay property of rings and book recommendation on intuition

My professor asks me to give an example about a local non-CM rings that are CM after modding out at any minimal prime. After a while failing to do so, I believe it is impossible since there exists a ...
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-1 votes
1 answer
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What does the primitive relations "betweenness" defined in Cayley–Klein model? What's the difference to the Euclidean geometry?

What does the primitive relations "betweenness" defined in Cayley–Klein model? What's the difference to the Euclidean geometry? In the link below, is $n$ between $k$ and $m$ in the Klein-Beltrami ...
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1 answer
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Linear Algebra, Geometric Representation of the Span of a Set of Vectors

Given the vectors: [4, 3, 3], [0, 1, 1], and [-1, 0, 0] The question: Is the vector [4, 4, 3] in the span of the set? I believe it is NOT, since putting the augmented matrix for this set in row-...
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1 vote
1 answer
87 views

What is the geometric interpretation of a separable space?

What is the geometric interpretation of a separable space? I know the definition of a separable space and I can give some examples.
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2 answers
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Draw solutions to complex number inequality $0 < \arg[(1-i)\overline z ] \le \frac \pi 4$ [closed]

I have to draw a picture of $$\{ z\in\mathbb{ C } ; 0 < \arg[ (1-i)\overline z ] \le \frac \pi 4 \}$$ I totally don't get it and i don't even know how to start solving this.
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1 vote
1 answer
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Geometric interpretation of this condition

In a Differential Equations context we have the following theorem: $\textbf{Theorem:}$ Let $D\subseteq\mathbb{R}^n$ be some open set and $f:D\to\mathbb{R}^n$ a $\mathscr{C}^1$ vector field. Then, for ...
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1 vote
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What is the geometric interpretation of Ito's Lemma?

I've been studying Ito's Lemma for 1D case with 1D Brownian Motion. Here's the lemma just to remember: $$dX_t=\big(\frac{\partial g}{\partial t}+\frac{1}{2}\frac{\partial^2g}{\partial x^2}|_{x=B_t}\...
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1 answer
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Intuitive interpretation of $\frac{\partial}{\partial \boldsymbol B}(\boldsymbol {B}^T \boldsymbol {AB})$

In my statistics notes, I'm given some "useful matrix algebra results". Two of those results are: $$\boldsymbol{A} \;\text{ is a } p \times p \text{ matrix}$$ $$\boldsymbol{B} \;\text{ is a } p \...
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Interpretation Reflection principle

The reflection principle says that for the Markov process $(B_t,\mathcal{F}_t,P_x)$ associated with Brownian Motion it is satisfied that $P_0(\text{max}_{s\leq t} B_s \geq a) =2P_0(B_t\geq a)$ ...
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Corollary of Projection onto a closed convex set and geometric interpretation

I need help with geometric interpretation of this theorem and with the corollary of the theorem: Theorem: projection onto a closed convex set Let $K \subset H$ be a nonempty closet convex set. ...
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1 vote
1 answer
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simplex $B^{-1} \cdot A_j$ tableau interpretation

In an iteration of the simplex tableau implementation, what is the interpretation of the columns $B^{-1} \cdot A_j$ underneath each variable $x_j$?
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Representation for circumferences intersection

I'm trying to write a formula that represents the intersections points for $n$ circumferences. All of these circumferences intersect them to each other. Is there a good representation to explicit this ...
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3 answers
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Proof and Geometric intuition of $u \leq 2\ln(1+u)$ for $u \in [0,1]$.

How can I prove the inequality $$u \leq 2\ln(1+u)$$ for $u \in [0,1]$. What is the geometric intuition behind this inequality?
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