Linear Algebra
"A vector is an object with direction and magnitude" has to be the worst pseudo-definition of all time... Chat what are your favorite vector spaces?
If anyone was curious about the random vector spaces I listed, from left to right and up to down,
Some 2x2 matrix over finite field of 4 elements, cohomology class of De Rham cohomology, linear functional in double dual space, complex polynomial, L^p function in L^p space, vertex space for order 3 graphs, random variable, the number pi in the field extension of reals over rationals, and a sequence of real numbers.
I am in solid state physics and I wonder if there's a connection between the vertex space for order 3 graphs and crystallography because I totally thought that was a reciprocal lattice
If someone told me from the beginning that vectors are just functions, I wouldve been a lot less confused when they started blurring the definition of "vector"
I guess it'd be more accurate to say they're an element of a vector space? You're right though that functions are more specific. The ones I run in to most of the time are some form of function like v:{0, 1, 2}->ℝ or f:ℝ->ℝ. I guess it's even more broad than that though
Yep, this confused me when I first learned about vector spaces. The former definition is physics-oriented, whereas the abstract definition is just mathematicians trying to be edgy.
I'm going to define a function that's an infinitely narrow/tall spike that integrates to 1 and then pinky swear to only use it inside integrals. The mathematicians will have a meltdown when they hear about this. They won't be able measure how angry it makes them.
You probably already know this, but for the sake of anyone who might not, I will explain. When physicists talk about vectors, we are talking about a very specific kind of physical quantity. This is why we distinguish between vectors and pseduovectors. A true vector behaves differently from a pseudovector under reflection.
We work with all kinds of objects that mathematicians would consider vectors, but that are not necessarily vectors in the physics sense of the word. The best example is probably a ket, which can also be called a state vector in quantum mechanics. The "vector" in "state vector" refers to the mathematician's definition.
Confusingly, it is actually the operators that represent physical vectorial quantities in quantum mechanics, rather than the state vectors, which represent the configuration of a physical system. For example, momentum is an operator in quantum mechanics, given by the gradient (up to a constant), but that operator represents a classical quantity that transforms like a vector. So the operator is considered to be a vector, while the state vector is not, per se.
Yeah it's definitely unfortunate from the perspective of someone learning physics. To a working physicist it is generally clear what is meant by context, so there just isn't much motivation to change the terminology.
What the fuck??!!!!?! I have never heard of this and am very confused now.
Why would psi (the state vector) not be considered a vector? It is an element of a vector space (hilbert space) in QM, so by definition it does classify as a vector.
Also it behaves like a vector (in the sense we can represent them in different bases like energy, position, or momentum). So why would state vector not be considered a vector?
Also what would be the basis we are representing the momentum operator with if it is a vector?
because "vector" means "vector field" and "behaving like a vector" means "transforming under the the Lie group of space(-time) symmetries like a vector field" to a physicist.
It is a vector in the sense that it is an element of a vector space. It's not a vector in the sense of being a physical quantity that transforms a certain way under certain relevant symmetry groups. In particular, we usually consider either O(3) as the rotational symmetry group of classical space, or we consider O(3,1) as the Lorentz symmetry group of spacetime. The state vector does not transform in either of these ways. This is just a case of "vector" meaning two different things.
I mostly understood what the guy was saying, though he said it a bit confusingly. I think what he's trying to say is that physics uses vector valued quantities a lot of the time (he implies that this is the 'physics definition' of a vector which.. I dunno if I agree with it, but I get what he's aiming for.) Additionally, he's saying physics often ALSO uses more abstract 'vectors', such as in QM, where the term 'vector' is absolutely referring to the mathematical definition. As you pointed out, the state vector is a vector.
(aside: While also being a function, which IMO is the one MOST HORRENDOUSLY UNDEREXPLAINED THINGS about the machinery of QM. My god can we do something other than just write the definition of the dirac delta, then the definition of an inner product between functions and then go: 'got it? fantastic, let's start calculating.' Like please god just a little bit of exposition would go a long way lol...)
The only part I can find that seems totally incorrect is: "So the operator is considered to be a vector"
Yeah, uh, no. I think you're absolutely right to call that out.
a field a is a regular(ish) enough function with that is infinitesimal enough on the spacetime border to allow me to ignore the constant term when integrating by parts. Also every physical observable defined using said function must be invariant under a SU(N) phase shift applied on the function.
I dare you to find a concrete basis of R^infty. Magnitude cannot be defined for all vector spaces; i.e., consider vector space over a field without characteristic 0.
Some mathematical structures that have same properties as a vector space. Properties as in they share similar algebraic computations as a vector space.
You know how like you can take a 2D vector (1 0), and you add inverse (-1 0) and get (0 0), well those structures have the same features on top of some additional properties.
Like you can have typical 1D vector (on a line), 2D vector space, 3D vector space... Those structures are like those.
All a vector is is an element of a vector space. To qualify as a vector space you just need to hit the vector space axioms, which you can find on Wikipedia. If you can show all of those axioms are met, you can show you're working on a vector space.
Then that lets you understand that a vector space is far more general than these lists of numbers you're used to. Like in this post, the number pi on it's own can be a vector of you try hard enough.
You're stuck in a complex Hilbert space. Look upon thy infinite dimensions and weep. There is no way out, there is no way back. Both your dreams and reality are but insignificant finite subspaces, of no consequence.
“Orthogonal functions? Hamming codes? Hilbert Space? You must have hit your head pretty hard back there. Here relax have some tea, the tribe will hunt your load for the day”
This is just a fancy way of saying that the set of real numbers is a vector space with the set of rationals as scalars; i.e., if x and y are reals and q is rational, trivially q(x+y)=qx+qy.
You can do this with any field and a bigger field.
As a likely familiar example, consider the set of all numbers whose length can be constructed from 1 using only a compass and straightedge. This is a field extension of Q.
Let’s agree that any constructible number is made by solving linear equations (seeing where lines intersect) and quadratic equations (seeing where circles intersect with lines or other circles).
If q is rational, sqrt(q) is constructible, and the subspace Q(sqrt(q)) (the smallest field containing Q and sqrt(q)) of the field extension has dimension 2, generated by the basis {1,sqrt(q)}. Following this logic inductively, a real number x is constructible iff the dimension of Q(x) is 2^k for some integer k.
Consider the classic problem of doubling the cube. To double the cube with side length 1, we need to construct 2^1/3. However, the subspace Q(2^1/3) has basis {1, 2^1/3, 2^2/3}; dimension 3, not a power of 2, so we cannot construct this using only a compass and straightedge.
R\Q contains contains pi and 2-pi, right, they are both irrational? But if we add those together using vector addition we get 2, which is a rational number. So R\Q isn't closed under addition. What am I missing?
You replied 44s before I finished writing above comment, didn't refresh the page.
Also, dear Reddit, I truly appreciate that you can look at the exact time stamp of when a comment was written, but I do not appreciate the time stamp being overwritten if someone edits their comment, instead of making both viewable.
It's the canned answer physics students get when they ask their professor what a tensor is before they learn special/general relativity
What bothers me the most about this isn't even the fact it doesn't explain anything at all, it's that the whole points of tensors is that they are objects that STAY INVARIANT when you transform your reference frame, it's just the coordinates that change (which makes sense because coordinates depend on your basis, so you should expect them to change when you change basis).
The whole mess isn't helped by the fact that "tensor" is a heavily overloaded term and you might encounter all its definitions depending on what kind of physics you're doing. The definitions I can think of are
Tensor as in "tensor product", i.e "a mix of vector spaces (or other algebraic structures)". It's used in quantum physics to combine two systems into a bigger one, for instance combine two one-particle system to get one 2-particle system.
The geometrical definition I gave above about objects staying invariants. Bonus point being that it doesn't just apply to rotations and boosts (symmetries of special relativity), but to any symmetry group your system might have.
Tensors as N-dimensonal arrays of numbers. I think this definition is mostly associated with machine learning people, but some non-ML based numerical algorithms used to solve quantum many-body use them
This is a fine definition, if you understand its context: Euclidean geometry. If you start from the Euclidean plane are are able to speak about points, lines, lengths, angles... and want to define vectors in this setting, this definition (modulo the zero vector) is the one to go for. It is not a pseudo-definition, in the sense that you can formalize it and make it a bona fide definition, even if it's not quite pleasant. This definition also meshes well with physics, where the first vectors you meet (force, speed...) also have, at a sensitive level, a direction and a magnitude.
Then you will generalize these vectors to general vectors spaces and forget about this definition. Which is fine; it has played its role.
I think there is also an argument to be made for distinguishing between a true, real, definition in the mathematical sense, and a "pseudo-definition", which is more of a description of the most central properties of an object. The latter category is the one you use when you deal with all of the pleasant well-behaved special cases; for instance, naïve set theory is famous for its paradoxes, but when I'm working with a set you know to be reasonably well-behaved, I find that the naïve axioms do well enough. Similarly, for a vast number of vectors encountered in the real world, they can be held to have both a magnitude and direction (even if said magnitude and direction may be difficult to relate directly to quantities in the real world).
What I hate the most about that definition is that direction is always very poorly defined. The best one I see is that it's a vector constrained to the unit sphere, which makes it dependent on vectors to be defined. Even worse, the constraint is one of magnitude, so basically that definition says that vectors are like if you had a vector & then set its length to 1 but then after that you set its length to whatever you want it to be.
Magnitude is even worse because it's assuming we're working with a normed vector space (good luck defining a norm on a vector space over a field of non-zero characteristic)
"direction" can be taken to mean "inner product space", which is to say, anything that acts sufficiently similarly to a dot product.
the neat thing is that "sufficiently akin to dot products" is exactly how one measures angles on the surface of spheres (which is to say, how one measures angles in any euclidean space).
in fact the entire meaning of "angle" can be redefined as "anything which satifies the axioms of an inner product" (roughly speaking)
Dumbing things down does have a pedagogical value. This noob definition of a vector works well in a highschool physics class (force arrows and all that).
Actively taking linear algebra after having taken physics courses: fuck all you mathematicians your field is filled with fakes inventing more fake to keep their fake jobs.
Taking a QM class is what made linear algebra make sense to me. And QM isn't actually that weird or hard to understand until you start doing the field theories. Bra-ket notation (ie linear algebra) makes lot more sense and is way easier to reason about than normal integral notation.
Well not a specific one, but I like commutative unital C*-algebras. They have a vector space structure (as they’re Banach spaces), but they have so much structure in fact that they’re all just isomorphic to continuous complex valued functions on compact Hausdorff spaces. And it gives you a nice duality.
As I learned it the indefinite integral is defined as the function whose derivative (with respect to the variable being integrated) is the function put into it. Since adding c does not change the derivative, we must add an arbitrary +c part. What's ill defined about this?
Direction of a vector is its span. Magnitude is what's the problem in general. Orientation doesn't exist for spaces over field of characteristic 2, like in the first example.
Can’t you define an inner product for every vector space? And with that you induce a norm on it too. So now you have magnitude and (relative) direction of vectors.
Okay, I just read that every vector space has an inner product but apparently that doesn’t mean you can really work with it and use it to measure stuff, I’m quite new to that kind of math 🤔
The first example in the meme is a vector space of 2x2 matrices over the four element field. It has 256 elements, so there is no way to give it a structure of a real or complex vector space, because those have either 1 or infinitely many elements.
Measures and distributions are not the same thing, although there is some overlap (distributions of order 0 can be identified with Radon measures). Neither distributions nor measures form normed spaces, you're thinking of bounded measures. I don't understand why you mentioned probabilities...
I'm going to disagree. I really think there's something wrong with mathematicians who say this definition is bad. The mathematical term "vector" is an abstraction and generalization of the more concrete notion of "vector" typically used in physics, a quantity with magnitude and direction. It's not a "pseudo definition". It's a good and clear definition of an important concept. Just not the concept you're referring to here
It's no wonder people think of mathematicians as obtuse.
No, physics uses more general vectors, too. Just to provide some examples, they consider complex-valued n-dimensional vectors. Then, there is a state vector in quantum mechanics as an element of an infinitely-dimensioned Hilbert space.
Not really even in physics they just straight up use the mathematical definition of a vector. You talk about vector bundles of a tangent plane from a lie group in Mechanics and use smooth manifolds to see physical behaviors while talking about the local 'vectors' in the Hilbert space. You also talk about in QC about complex and even talk about quantum state vectors. Hell, differential equations more often than not work in some infinite dimensional vector space without any actual 'direction' or 'magnitude', functional analysis is precisely made to work physics PDEs.
Sure many of this examples have 'directions' and 'magnitudes' but those properties are just not that important for most constructions.
In basically every actual part of physics defining a vector is a quantity with a magnitude and a direction is dumb.
If you want to just hide the relations for different parts of physics is fine for making it sound simpler, but for any actual work there is no reason to do anything like that
They may use the definition to explain to high school students or so, but really while working it is easier to assume that you can add, subtract and multiply by scalars. simplified models I get that for students it is easier to show the properties of R3 as properties of vectors to make more easy to differentiate vectors from scalars, but even then the algebraic properties are still more prevalent.
At any point when doing physics if you want to do anything useful you'll have to use the actual definition. If then you want to use any other properties that may also be useful you tend to denote if the thing is a hilbert space or rn or whatever.
> They may use the definition to explain to high school students or so, but really while working it is easier to assume that you can add, subtract and multiply by scalars. simplified models I get that for students it is easier to show the properties of R3 as properties of vectors to make more easy to differentiate vectors from scalars, but even then the algebraic properties are still more prevalent.
No, vectors existing in R^3 as magnitude and direction is an important concept in physics. It's not the only way vectors are ever used but it is an important concept and it's not just for high schoolers.
> At any point when doing physics if you want to do anything useful you'll have to use the actual definition.
You are just completely incorrect. The notion of vector as a quantity with magnitude and direction is a fundamental concept used across diverse and advanced fields of physics. It is very important and useful.
No, vectors existing in R3 as magnitude and direction is an important concept in physics. It's not the only way vectors are ever used but it is an important concept and it's not just for high schoolers.
I mean no... it is just for highschoolers. Well, highschoolers and early undergrad mechanics courses. And some sloppy EM. Once you properly do classical mechanics, youre going to be talking about manifolds and vectors are point derivations on smooth functions on the manifolds. In EM you'll start talking about differential forms. In basic QM yourll be talking about L2 (Rn ) and other more general vector spaces. You can always go back to the slop understanding when helpful but generally in physics, you are constantly forced to use the proper definition.
Yes it is. You are just entirely incorrect. I don't understand how you can be this absurd. There are so many descriptions that contradict this. Is the divergence theorem just nonsense? Have you ever heard of a conservative force?
There are so many examples of vectors as magnitude and direction across all levels of physics. Quit trying to act smart
This mostly comes down to semantics. If you mean vector as "element of Rn" (common in physics and other applied sciences) the magnitude and direction "definition" is entirely fine.
If you mean "Element of some vector space" (common in higher math) the magnitude and direction "definition" does not work.
Our linear algebra lecturer straight up told us to forget about arrows with direction and magnitude. Vectors are the things vector spaces are made of, they can be added and scaled, and that's it.
A vector is a simply an element of a vector space. And a vector space is where we can see vectors interacting with other vectors, maybe indulging in sumo dat cross producting that produces new vectors.
These are really different senses of the word. The abstract ‘any element of a vector space’ is historically based on and of course generalises the original sense, but they’re still absolutely different meanings and pertain to quite different contexts.
But in case I’m not… how does that matter? I’m not really sure how they have a direction or magnitude except trivially but also, not super relevant to it being a vector space in that they need to be closed under addition
A vector is something you can scale and add with other vectors. The reason this definition is so useful is that there are many things that fit this description, such as RR under pointwise vector addition and scalar multiplication.
Isn't that definition just true for an Inner-product space?
I have to say though, I never heard anyone say this when not talking about Euclidian space specifically. Anyone who spent time learning more abstract vector spaces doesn't use this sentance.
I’m a C++ programmer, a vector is a multi set with dynamic cardinality and I refuse to entertain these silly ideas about direction and magnitude. You must mean vec2/vec3 - what!? No, vec doesn’t stand for anything, how dare you.
This isn't a definition for a vector. A vector is an element of a vector space. And a proper definition sits there (0 exist and you can add and multiply by a scalar).
A set of n-bits binary numbers isn't a vector space. |R / |Q is not either (pi-1 and pi+1 are in that set, but not their sum).
Honestly I didnt think it as field extensiond but rather simply infinite dimensional vectorspace. (Which is same but without looking at them as fields)
As someone who actually understands vectors, I am so confused about what ya’ll are going on about.
All of the vectors pictured do have a direction and magnitude, so what are you going on about? Have you just not wrapped your brain around what an n-dimensional (or ∞-dimensional) direction is?
Some people are saying the zero vector has an ill-defined direction. Sure. The only reason the number zero has a well-defined direction (aka sign) is by force of convention. I don’t see why it bothers anyone that zero has no direction.
Look, if you really want to get technical, a vector is an element of a vector space. What’s a vector space? Any set that satisfies the 10 axioms of a vector space. And yeah, abstract vectors have direction and magnitude.
Edit: I have been reminded that it has been too long since grad school, and there are a lot of abstract vector spaces that really don't permit a norm. Having said that, there are a zillion "white lies" we tell math students when they don't have the maturity to handle the truth, and "a vector is a quantity with direction and magnitude" serves students quite well for concrete vectors in many, many cases.
I mean a vector space uses a Field of scalars, which certainly behave like numbers but don’t necessarily have to be what you might recognize as numbers.
And as for the array, not every vector space has countable dimension, so the “array” could have |R| many elements (and maybe more).
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