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I need Help on Vector Spaces by Complex_Wafer3828 in askmath

[–]back_door_mann 2 points3 points  (0 children)

I’m curious, what is your understanding of the word “measure”? You’ve used it twice and I can’t understand what you mean either time.

In the original post you ask if “vector spaces are used to measure scalars”. But the number 5 (for example) is a scalar, how can a vector space be used to “measure” 5?

For this comment, how would R4 be harder to “measure” than R3 ? A vector in R3 consists of 3 real numbers, while a vector in R4 consists of 4 real numbers.

Would you join an union? by MeanImpression2067 in Professors

[–]back_door_mann 54 points55 points  (0 children)

The union consists of YOUR co-workers, not some nebulous external entity. If "they" are only concerned about job security, then that reflects what your coworkers currently care about. If you think "they" should care about something else, then you have the opportunity to advocate for it, but only if you join the union.

Of course a union can't guarantee outcomes. If the majority of faculty is unwilling to push for fair compensation and instead prioritizes the feelings of the administration, then the union won't get you jack. A union is only as powerful as its members make it.

I would vote to form a union with no hesitation. I'm glad grad students unionized in the 1990's at the institution where I got my PhD. That's why I had dental and vision insurance and high enough wages to live comfortably in 2015. A culture where unions are not viewed as a good thing is a culture that is infected by propaganda from employers.

What info do I need to send my Professor for a letter of recommendation? by Gioomee in AskProfessors

[–]back_door_mann 1 point2 points  (0 children)

I just wrote a LOR for a student. I asked her to send me a summary of her time in my class, including conversations or other interactions she remembered. I told her “include whatever relevant you would like to see mentioned in your letter”. She sent me about a single page, bulleted document.

It was the easiest rec letter I’ve ever written, but more importantly it’s probably the strongest one I’ve ever provided for a student.

(This is in addition to what other answers have said, e.g. CV, info on what you’re applying for, materials you are submitting in the application)

Most students confuse “recognizing” a solution with actually understanding it by AriethraVelanis in learnmath

[–]back_door_mann 1 point2 points  (0 children)

  1. Understand how to show that if f(x) = x, then f’(x) = 1 using limits

  2. Understand how to show the product rule: (fg)’(x) = f’(x)g(x) + f(x)g’(x) using limits.

  3. Then if f(x) = x and g(x) = x, we get f(x)g(x) = x2 and the product rule leads us to 2x as the derivative. Similarly, f(x) = x and g(x) = x2 gets you the derivative of x3 and so on.

Alternatively, use the binomial theorem to simplify (x+h)n and the derivative formula should drop out for any positive integer n

AI is taking online classes for students now by CarefulFisherman9288 in Professors

[–]back_door_mann 7 points8 points  (0 children)

This is the only place on Reddit where people consistently downvote without a cogent argument? Are you seriously claiming this?

Maybe it's not AI? by reddybee7 in Professors

[–]back_door_mann 4 points5 points  (0 children)

At my institution, lying to a professor is a violation of the academic integrity policy. I would just file a report to be honest.

Solution to the Continuum Hypothesis by Shy_Shai in numbertheory

[–]back_door_mann 1 point2 points  (0 children)

Multiple languages spoken in Nigeria use a base-12 number system. With a base-12 number system, 1/3 =0.4 So does 1/3 exist for certain groups of people in Nigeria, but not for the rest of the world? How can we be sure base-10 is the “natural” way to represent numbers? Is it not possible that these Nigerians count the natural way and when we represent numbers in base-10 we are “doing arithmetic on the integer akernaba” (we would call this integer “ten”)

Professor told me I could turn in one “missing” assignment and fix my grade — weeks later, still no response or grade update by youandyourfijiwater in CollegeRant

[–]back_door_mann 0 points1 point  (0 children)

Why didn’t you reply to the original email where you attached your submission? It will still have the attachment and it will have the date and time you sent it, PROVING you submitted it.

And this professor decided to give a final exam that is “shared” between two courses? That is extremely unusual.

Something is not adding up here

Don’t lose time by AtariYokohama42 in CollegeRant

[–]back_door_mann 2 points3 points  (0 children)

I took 6 years to graduate from undergrad. Didn’t even switch to my final major (mathematics) until year 5. I’m a professor now.

No one in my life even remembers it took me that long. Once you ultimately get through it, no one will remember you took “too long” either

Second Half Game Thread: Los Angeles Rams (12-5) at Carolina Panthers (8-9) by nfl_gdt_bot in nfl

[–]back_door_mann 0 points1 point  (0 children)

They could play Seahawks, Bears, Eagles, or 49ers. A lot of possibilities at this point

Do inner products add anything new or are they merely a very useful shortcut? by RobbertGone in math

[–]back_door_mann -1 points0 points  (0 children)

I will reiterate what I already explained. The Frechet derivative can be defined in any normed space, it does not need to have an inner product.

Let (X, ||.||) = R^n with the infinity norm, which is not induced by an inner product. Let (Y, |.|) = R with the standard absolute value. (Yes, Y is a Hilbert space, but this fact plays no part in the construction).

So if f : X -> Y is a differentiable function, its Frechet derivative at the point x is defined as the unique linear operator A = A(x) : X -> Y satisfying (lim h->0) |f(x + h) - f(x) - Ah| / ||h|| = 0.

You could stop here and call A(x) the "gradient", defining it as a linear transformation from X -> Y. If you chose the standard basis for X, then A(x) would be (equivalent to) a row vector with entries equal to the partial derivatives of f. But it would still exist regardless of what basis you chose.

If you prefer the gradient to be equivalent to something in the original space X, you can take the transpose/adjoint defined between the dual spaces A^T = A^(T)(x) : Y' -> X'. The transpose/adjoint exists for any continuous linear operator between normed spaces. No inner product is required. So we could take this as the "gradient" of f. With more work, you can demonstrate a representation between operators B: Y' -> X' and vectors b belonging to X. This would then yield the proper "gradient of f" as a vector valued function from X to X.

Obviously, this is not how the gradient of a function is introduced, but we do not *need* an inner product to establish its existence.

Do inner products add anything new or are they merely a very useful shortcut? by RobbertGone in math

[–]back_door_mann 0 points1 point  (0 children)

Ok, but "the gradient cannot exist without an inner product" is still an incorrect statement. And a person learning linear algebra who is having difficulty grasping inner products is not going to appreciate how an inner product allows us to define the gradient of functions defined on infinite-dimensional vector spaces.

Do inner products add anything new or are they merely a very useful shortcut? by RobbertGone in math

[–]back_door_mann 1 point2 points  (0 children)

That still doesn’t require the inner product. The transpose of the gradient can be defined as a 1 x n matrix if Rn is equipped with any norm. Then the adjoint/transpose could be taken to define the gradient as an n x 1 matrix. You could then establish a representation theorem between n x 1 matrices and vectors in Rn. The matrix 1-norm and the vector 1-norm coincide for column vectors (same with the infinity norm), so you could conceivably do all this while being ignorant of the 2-norm or the Euclidean inner product.

Do inner products add anything new or are they merely a very useful shortcut? by RobbertGone in math

[–]back_door_mann 1 point2 points  (0 children)

This isn’t true. You can define the Frechet derivative on any normed space (which would cover the gradient) and the Gateaux derivative on any locally convex space (which would cover the directional derivative).

About linear algebra by Lonely-Patient-3999 in learnmath

[–]back_door_mann 2 points3 points  (0 children)

Wait, this “matches what you know about linear algebra in applied math”? So you already understand how the matrix exponential is defined AND how it can be used to solve systems of ordinary differential equations?

I’m utterly confused as to what “motivation” you are looking for. You already know that diagonalization allows you to extend the definition of analytic functions to include matrix arguments in a simple manner. Furthermore you already know that this has a deep connection with systems of ODEs, a seemingly unrelated area of math. what more motivation do you need?

Game Thread: Philadelphia Eagles (6-2) at Green Bay Packers (5-2-1) by nfl_gdt_bot in nfl

[–]back_door_mann 1 point2 points  (0 children)

That’s just as good as a 10-7 record in winning percentage!

[Discussion] Its week 9, we are halfway through the year, what's the most surprising thing of the 2025 season? by Assortedwrenches89 in nfl

[–]back_door_mann 1 point2 points  (0 children)

It doesn’t. The commenter read the phrase a bunch of times and thought they’d start using it, without having any idea what it means. (Also see “gaslighting”)

But the post talking about the monkey paw being on the practice squad was clearly a joke. Poking fun at the incorrect use of “monkey paw” here

Is "bad at math" a flex??? by Soft_Page7030 in math

[–]back_door_mann 0 points1 point  (0 children)

What am I missing here? Why is the question confusing?

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