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Questions tagged [intuition]

Mathematical intuition is the instinctive impression regarding mathematical ideas which originate naturally without regard to formal mathematical proofs. It may or may not stem from a cognitive rational process.

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Intuitive conditions to recognize a non diagonalizable matrix.

I know that to find out whether or not a matrix is diagonalizable you have to study eigenvectors and eigenvalues and the space they generate. I also know some shortcuts to find out if it is ...
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Intuition behind affine subsets?

I am working through Axler's "Linear Algebra Done Right" and I am having trouble intuiting some of the meaning behind affine subsets. According to 2 exercises in the book we have that (1) A subset $...
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Need help with a simple example where it's not clear that the gradient is in direction of “steepest ascent”

Say I am on a point $(x^*,y^*)$ of a function $f(x,y)$ where the function value increases if I go a very small step in any positive direction (i.e. in the direction of a vector where the coordinates $...
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Total and partial derivative terminology in scalar- and vector-valued functions of scalars/vectors

I have the following terminology questions which are often not well addressed in an undergraduate multivariable calculus course. While the questions are long, I expect their answers will be short. As ...
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Possible non geometric intuition for the mean value theorem? [duplicate]

Is it possible to "see" the mean value theorem when one is only thinking of numerical values without visualizing a graph? Perhaps through a real world problem?
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Is my definition of discontinuity correct? [duplicate]

I think I have made a rigorous and intuitive definition of discontinuity. Definition: For a function $f(x)$ from $\mathbf{R}$ to $\mathbf{R}$, there is a discontinuity of $a$ at $x_0 \in \mathbf{R}$...
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Intuition on the direction of steepest ascent always being orthogonal to the level set of the function

Thanks for reading. THE QUESTION: Convince me that when on the surface of a smooth hill, the $(x,y)$ direction I should take a tiny step in such that my current height doesn't change is always ...
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I want to know what was the motivation/intuition behind weierstrass function?

I know that it came as an example of a function that is continuous everywhere but differentiable nowhere.But how did the idea of such an unusual construction came?
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Visualize the proof of $\mathcal{H} = M \oplus M^{\perp}$, where $\mathcal{H}$ Hilbert space, $M$ closed subspace

Lemma. Let $M$ be a close subspace of a Hilbert space $\mathcal{H}$. Then $\mathcal{H} = M \oplus M^{\perp}$, where $\oplus$ denotes the orthogonal direct sum. Proof. Let $x \in \mathcal{H}$. We aim ...
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Intuition behind “Transpose Matrix”

I have come across the differential $\frac{\partial \bf{w}}{\partial \bf{w}^T}$ many times now, and I notice that it is equivalent to the transpose operator. That is, if we have something of the form $...
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Definition of the Fibre of a Sheaf

I'm working through an exercise and have a few questions about the following construction. If anyone thinks they should be split into separate posts please let me know, but they seem related and I'm ...
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How do I use the Iverson Bracket notation to calculate the expectation of geometric distribution?

I learned from this answer that I can use the Iverson bracket notation to get the expectation of the geometric distribution. The theory behind it seems unfathomable to me now and I just want to intuit ...
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Intuitive perspective of eigenvalues and rank of a matrix

Assuming a matrix $A$, $n\times n$, with $n$ non-repeated and non-zero eigenvalues; If we calculate the matrix $A-\lambda I$ for one of its $n$ eigenvalues, we see that its rank has been decreased by ...
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1answer
36 views

The idea of a bundle chart and bundle atlas.

The definition of a bundle charts and bundle atlas is rather obscure in my opinion. Is it fair to say that: 1) the purpose of a bundle chart is to give coordinates to each tangent space? 2) the ...
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Solving mathematical economics problem

kindly I'm stuck in this problem instead of many attempts through net present value and other discounted cash flow methods, some one could give me a detailed information and answers on this problem : (...
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66 views

Why is multiplication a commutative operation?

This trivial question is all about reasoning (intuition) and obviously not proving. I know $a\cdot b = b\cdot a$ from very early school years and it's considered intuitive. A simple proof is by taking ...
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Elementary reason for no closed form for the sum of first n terms of harmonic progression

I tried a hard for getting a concise formula for the sum of first n terms of a harmonic progression. After a couple of years, I found out that there exist formulae in the continuous case (say log n)....
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Why is it so important in the method of steepest descent that we follow the steepest descent contour that passes through a saddle point?

In the method of steepest descent, we approximate to leading order an integral of the form $$I(x) = \int_C f(t) \exp(x \phi(t)) dt$$ as $x \to \infty$ by observing that if we write $t = \xi + i \eta$ ...
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In random variables, how come we can add up the variances but not the standard deviations?

I was watching a video from Khan Academy and he mentioned that we can add up the variances of random variables but not their standard deviations. Can someone help me find the intuition on why?
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Why are there two formulas for variance of random variables?

I'm using an introductory statistics textbook and it mentioned this: Definition: If $X$ is a random variable with mean $E(X) = \mu$, then the variance of $X$ is defined by $Var(X) = E((X−\mu)^2)$....
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Understanding this proof about linearity of expectation

I'm studying introductory statistis and I'm having a difficult time understanding linearity of expectation. I found this proof from the website Brilliant: We'll explicitly prove this theorem for ...
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Intuition and understanding the addition and subtraction rules of expected values

I'm a beginner learning statistics through Khan Academy and it mentioned this without any explanation on why it "works"/the intuition behind it: $\mathbb{E}(X+Y) = \mu_{x+y} = \mu_x + \mu_y$ $\...
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What is the intuition behind uniform continuity?

There’s another post asking for the motivation behind uniform continuity. I’m not a huge fan of it since the top-rated comment spoke about local and global interactions of information, and frankly I ...
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1answer
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Integral with $dF(x)$ instead $dx$ notation

What does it mean when we write $\int\limits_a^b f(x)dg(x)$ and how to translate it into an integral with $dx$ instead? I get intuitively that the small rate of change along the $x$ axis is no longer ...
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Why do we care about complex inner products and what do they mean? [duplicate]

If I have two vectors from $\mathbb{R}^q$ then their inner product gives the length of the projection of one on them onto the other multiplied by the other's length. I have searched but couldn't find ...
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Solving the BVP using finite differences and intuition regarding the form of the numerical solution?

Suppose we have $u'' + u' =0$ where $x \in (0,1)$ with the boundary conditions $u(0) =0, u(1) =1$. We consider the BVP finite difference approximation $\frac{u_{j+1} -2u_{j} + u_{j-1}}{h^2} + \...
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103 views

Tautological one form assigns a numerical value to the momentum $p$ for each velocity?

In the wiki, it is written that : the tautological one-form assigns a numerical value to the momentum $p$ for each velocity $\dot {q}$, and more: it does so such that they point "in the same ...
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78 views

Example for Hilbert quote

Hilbert famously said The art of doing mathematics consists in finding that special case which contains all the germs of generality. Can you give an example of a situation in mathematics where ...
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1answer
42 views

Connection between fundamental group of circle and division by zero

Going through some of the applications of the proof that $\pi_{1}(S^{1}) \cong \mathbb{Z}$ in Hatcher, I have noticed that almost all of these arguments rely on a contradiction whose assumption ...
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1answer
24 views

Difference between a binomial experiment and a Bernoulli Trial?

I'm studying in an introductory statistics textbook and it's confusing me when it mentioned this: "Any experiment that has characteristics two and three and where n= 1 is called a Bernoulli Trial(...
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Intuition behind why a binomial experiment has to have a fixed number trials

I'm studying an introductory statistics textbook and it mentioned this: "There are three characteristics of a binomial experiment: There are a fixed number of trials. Think of trials as repetitions ...
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Tossing Coin Expected Value Against Intuition

I have the following two coin toss games: Game 1: A and B tosses a coin. At first the coin is unbiased. Through the game, if heads comes A wins and game stops. If tail comes, the coin is swapped with ...
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Why is the Orbit Stabiliser Theorem intuitively true?

Why is the OST intuitively true? (Specially for the finite groups but also infinite groups) I understand the proof and the steps, but it is not obvious to me like let’s say Intermediate Value theorem.
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How to prove $\frac{\partial}{\partial x^r} (x^r) = 1$ where $(U, (x^1, …, x^n))$ is a local chart of a smooth manifold.

Let $M$ be a smooth manifold. I am getting lost in the notations. Could someone please explain me how to prove $\frac{\partial}{\partial x^r} (x^r) = 1$ where $(U, \phi = (x^1, ..., x^n))$ is a local ...
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Is the ideal $I=(X+Y,X-Y)$ in the polynomial ring $\mathbb{C}[X,Y]$ a prime ideal?

Is the ideal $I=(X+Y,X-Y)$ in the polynomial ring $\mathbb{C}[X,Y]$ a prime ideal? Justify your answer. I am a bit hesitant about asking this here. The question is not "How to Solve This Problem". ...
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Rigidity Theorem, meaning.

What does it mean to say that a theorem is a rigidity theorem? I'm reading a book (Lipman Bers, a Life in Mathematics) that says for example that Torelli's theorem is a rigidity theorem. So what ...
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Intuition behind the probability of an event and another event's complement

I don't get the intuition when my introductory statistics textbooks mentions this, can someone please explain it to me?: P(D and B`) = P(D) - P(D and B)
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Visual Intuition: Gaussians closed under addition

I'm trying to develop some intuition for the fact that the family of Gaussian distributions is closed under addition. I.e. if $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$, then $Y = \sum_iX_i$ is also ...
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1answer
70 views

Power Set and Empty Set question

I have a question regarding the set of functions resulting from a set raised to a power. I think I have part of the understanding correct, however I'm having trouble interpreting $Y^{\emptyset}$. I ...
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1answer
76 views

Why is the definition of inductive set well defined?

I've been studying from Enderton's Mathematical Introduction to Logic in which he defines an inductive set as follows: To simplify our discussion, we will consider an initial set $B \subseteq U$ ...
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31 views

What it means that multiplication is stretching and addition is sliding the number line?

I have read these definitions but I am not sure that I understand them. Could someone give the intuition behind these definitions?
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The relation between the F1 score and the combined work formula?

I am wondering why we should use the harmonic mean(as stated in Wikipedia: "the traditional F-measure or balanced F-score (F1 score) is the harmonic mean of precision and recall") as the F1 measure, ...
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2answers
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Consider a function $f(x,y,z(x,v))$. Does $\frac{\partial f}{\partial z}$ hold $x$ and $v$ constant?

where $x,y,v$ are independent variables. Basically, I am confused with the fact that a partial derivatives holds all variables constant, other than the variable with which we are taking the ...
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What's the geometric interpretation of the square root of a matrix?

Question: If I have a matrix $A$, I know that its square root is a matrix that has the same eigenvectors as $A$ but its eigenvalues are the square roots of the eigenvalues of $A$. What does this ...
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66 views

Intuition behind proof of Schwarz's lemma

There is the very well known proof of Schwarz's lemma in complex analysis. When I read it I feel like the answers described here. I'm not sure how I would motivate and explain why one should expect ...
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1answer
26 views

Intuition behind independent events in introductory statistics

I'm learning through an introductory statistics textbook, and I just can't get the intuition when the textbook mentioned this for independent events $A$ and $B$: $\bullet$ P(A|B) = P(A) $\bullet$ P(...
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5answers
44 views

Taylor Series - why that specific form?

Something that has bugged me since university. Why does the Taylor Series have that specific form? For example there is a division by n! - why not (say) (n^2)! How does one get to the Taylor Series? ...
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1answer
32 views

Intuition of finding the formula of Laplace's expansions of the Determinant?

How did Laplace find the formula $\left |A \right |=\sum_{i=1}^{n}(-1)^{i+j}(A)_{ij}M_{ij}$? What is the intuition of the evalution of this formula? Note: I'm not asking for proof that the formula ...
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1answer
50 views

Connections and metrics on Riemannian manifolds

In a lecture on connections it was claimed that a manifold "attains a shape" when it is equipped with a connection while there were no mentions of a metric. Is it correct to intuitively think that ...
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Describe the Bidual of Real Polynomials

Trying to understand (bi)duals of infinite-dimension vector spaces, I stumbled over the very concrete example of $\mathbb{R}[X]$, the (formal) polynomials over $\mathbb{R}$ or, equivalently, the space ...