Vector Spaces Puns

So I noticed kind of two major patterns for how Linear Algebra is thought.

The traditional university method seems to be:

• define a matrix as an abstract mathematical tool
• what are elementary row operations
• modeling systems of equations and solving via row operation
• Determinants, Inverse Matrices, Factorization, etc
• (later in the semester) Vector Spaces

A method I've seem more common outside of university math class is the more geometric approach:

• Introduce the geometric concept of vector spaces
• basic vector operations
• basis vectors and intro to vector space
• Introduce the concept of a matrix as a transformation of a vector space

Now to me it seems like the the second way is very obviously superior. As it teaches the intuition for what Linear Algebra is, and then derives the idea of a matrix from this geometric representation. Also, all the elementary matrix operation stuff to me is not super interesting, as most matrix solving is not something I would ever want to do by hand.

What do you all think? Is this preference just a result of personal bias? Why does universities typically tend towards the first method?

I am trying to get at the essence of a vector space, but each answer uses examples involving (real) numbers. Or functions of real numbers.

I want to see if we can construct a system, using the concepts of vectors in a vector space, and not have it directly involve numbers at all. Is this possible? Basically, can vectors/vector-spaces be used to model things outside of numbers?

I get that a vector space requires satisfying some 8 axioms, or as Wikipedia says, "a vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below."

Can you construct a system of vectors in a vector space which satisfies these properties/axioms, and yet doesn't directly revolve entirely around numbers (so isn't using the real numbers directly, or functions of real numbers, or matrices of real numbers, etc.). Is it possible?

My starting of an example would be something along the lines of... Say for example we try to make the vectors be molecules. Can it be done? In this way, they will not be directly related to numbers.

• Associativity of vector addition (example, elements are acid/base/water, acid + base = neutral element)
• Commutativity of vector addition (acid + base = base + acid)
• Identity element of vector addition (v + 0 = v, maybe perhaps an unreactive molecule)
• Inverse elements of vector addition (v + (−v) = 0, acid + its opposing base = water)
• Compatibility of scalar multiplication with field multiplication (a(bv) = (ab)v, dunno)
• Identity element of scalar multiplication (would have to think harder how to fit the rest into the mold...)
• Distributivity of scalar multiplication with respect to vector addition ...
• Distributivity of scalar multiplication with respect to field addition ...

Maybe instead of molecules, we use some other objects like light waves or colors, or human beings, etc. Can vector spaces have as their "vectors" arrays of arbitrary non-numerically-related objects like these? If so, what is a complete example?

At all costs, please don't write about something to do with numbers, the real numbers, or anything directly referencing the real numbers.

If it can't be done, why not? If it can be done, what is an example, and if you can, what is a practical application of vectors as NOT-numbers in the real-w

Let V be a vector space over the complex numbers that has a countable basis, and suppose A : V → V is a linear transformation.

Prove that there must exist a complex number λ such that the linear transformation A - λI is not invertible, where I is the identity transformation.

Trying to figure out a battery that will actually fit in this very small space. Have a lipo that fits but it runs Tamya connector which does not fit in the vector (not sure why its even included)

So, is there an on market lipo battery that ships with deans connectors, I've never messed with a Sauder and frankly I'm not wanting to start unless I have to, so I need a preferably 11.1 lipo that both comes with deans and fits the krytac vector

I'm trying to show that the set of converging sequences in R is separable and I have found a countable set that has dense span (when spanning over R). Obviously, if the span over Q is also dense, I am done. Does Q's density in R imply my statement? Many thanks in advance

I am confused. I just don't see the relationship between field theory and vector spaces or linear algebra. I went down a Google rabbit hole, but everything I found assumed there was an obvious connection. Any help or resources will be highly appreciated.

https://i.imgur.com/IpadnAD.jpg

This is due tomorrow and I have no idea if what I think I should do is correct (Edit: it isn't, I was thinking about another type of problem). If you know how to solve these, please link an image with the procedure? Thanks in advance...

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this is continous action space since action is a real number or discrete as number of elements (100 here ) are fixed ?

Say I have a vector space V that's a finite direct sum of spaces W_i. Let R = End(V). Let G be a group (finite or infinite) and consider the group ring R[G] = End(V)[G].

How does R[G] compare to:

1. the new group ring with R' = direct sum of End(W_i)
2. Direct sum of group rings End(W_i)[G] as free modules (if even possible)?

I encountered both words frequently. They seem to refer to same things. I am so confused. Anybody care to help clarify? Thanks.

Every week we get new topic to study in our college, and then at the end of 4 weeks we have to give a test based on those 4 weeks.

first 2 weeks were Determinants and Matrix.

I really struggled with those, but somehow managed to learnn the things, at least the bare necessity so i can answer the exam questions. There are topics like finding inverse of a matrix or finding the determinant or solving a bunch of linear equations using matrix etc. And I can do some of those like I am somewhat comfortable with it. I mean, as comfortable as you can be in just two weeks.

But that's not the problem. The problem is week three, where we now are being taught vector spaces, which is a topic which I just can't get my head around. None of it makes any sense to me and I have tried all the not all the like quite a few YouTube materials and. I am still very confused. I was hoping someone out there could refer me to some material which is in a very elementary or basic way to explain vector spaces and answer questions. A sample of the kind of questions that we have is that we are given sets and then we have to identify whether it forms a vector or not, whether it has a closed under addition or not, or whether it forms a vector space or not. There are questions such as choose the correct set of options, a matrix of order one by three is a vector or not, or something like that, I don't know. Then there are questions about linear dependent or linear independent. We are given sets like a set of 1234567 at 9 and identify it is independent. Dependent. Then, given a set of variables like a 1B1C2V2C3D3 etc, tell whether the linear equations has a unique solution or not, or whether the coefficient matrix is invertible or not. Weather subset of another given set is a. Linear dependent or linear independent? Linear combination.

The above paragraph I just randomly read out certain parts of the set of questions which we have been given to solve. None of those make any sense to me, but perhaps those of you who are experts can point me in a proper direction as to what, where and how to prepare this topic. Next paragraph. Change line.

Please help me. I have this examination on 30th January. If I could learn some of the concepts before that, it would really help me. Of course, I'm not solely depending upon this Reddit. This is just one of the many ways I am employing to. Solve my dilemma.

I've used Geometric Algebra (1) to rewrite Quantum Mechanics in a possibly new way. It emerged that if you use a way such that probability is calculated from an inner product of state multivectors, then a spin-1/2 particle reveals that space vectors are bivectors. Interesting...

The equations are

X=1/2*(a d + b c)

Y=1/2*(a b + c d)

Z=1/2*(b d - a c)

and you can derive space rotation in this GA formulation.

I've uploaded a derivation to Github.

Can you think of an interpretation why that is?

Also this may provide an alternative way to apply geometric algebra to quantum mechanics.

(1) Geometric algebra introduces an associative, non-commutative product between vectors such that for orthonormal vectors you have e_i^2=1 and they all anti-commute amongst each other.

EDIT: I've simplified and extended the write-up. Check it out for updates!

I am reading descriptions like these regarding observables in QM:

>"The eigenvectors of a von Neumann observable form an orthonormal basis for the Hilbert space, and each possible outcome of that measurement corresponds to one of the vectors comprising the basis."

I have some very basic questions, to gain a more intuitive understanding:

1. If you have an exactly 33-dimensional Hilbert Space, for example, then for any Hermitian operator (corresponding to some observable), there will always be exactly 33 eigenvectors (and they will all be orthogonal to each other)?

2. We care about the 33 orthogonal eigenvectors that get either scaled up or squished down to a length of 1 ("orthonormal"). The eigenvalue of some vector—i.e., the real scaling factor of this vector from before vs. after the operator—is the value of the measurement, if in fact the system ends up collapsing into that state vector?

3. When the measurement is made, is one of these 33 eigenvectors chosen? That is, the system "collapses" into one of these eigenvectors? And the eigenvalue is the actual value of the measurement?

Please let me know if I'm on the right track or wildly off! Thanks in advance!

so they basically told me that some functions live in a function space and that this is an infinite vector space. doe this means that there are an infinite amout of basis vectors to represent a function? im a bit confused can someone help me out

Let me see if what I got so far is correct:

A vector space is a set of vectors where:

• It has a zero vector

• linearly combining these vectors results in a vector that is also in the vector space.

A subspace of vector space V is a subset of vectors of V that:

• contains the zero vector in V

• linearly combining the vectors in this subspace results in a vector that is also in this same subspace.

a vector basis of a space(or subspace) is the set of linearly independent vectors of this space(or subspace). The total of vectors in this set must be the same of the total components of these vectors. So for a basis of a space of 2D vectors there must be two linearly independent vectors.

A set of vectors span R^n when this set has vectors with n components and all of them are linearly independent.

What did I miss in here?

Geometry is an art derived from the planet itself. It is what was studied by all civilizations and existed since the beginning of society. Primary and secondary education only focus on Euclidian geometry and often limits analysis to 2 dimension. For the masses, that is all that’s required.

For people seeking a superior understanding of this intricate system;

For people who want to journey beyond the plane;

For people who want to see the possibilities beyond their imagination;

For people who want a vector to guide them into a new dimension;

------------------—-

They need to shift their geometric cognition.

They need to supplement their synthetic geometry with a system of coordinates that allows for this understanding {3D Cartesian, Cylindrical, and Spherical Coordinates}

They need to embrace a perspective where position and orientation is defined by vectors {position, normal, and directional vectors}

They need to wire their minds to accept the temporal in addition to the spatial {parametric}.

They need to project the 3D world they feel onto the 2D world they see.

—————————

At the same time, there are those who only desire to live in the finite world.

All of calculus relies on the limit, a human construct, a weak attempt at trying to touch infinity.

Calculus attempts to create logical lines from free-flowing curves {derivative}, yet those creations cannot be constructed by the hands of humans. >!You can use compass to make a tangent to circle but not other curves.!<

—————————

Most universities entangle the infinitesimal nature of calculus with the horizon of the vector space into a course called Multivariable Calculus.

Though a powerful weapon, there are those who do not wish to wield it.

Those who want travel across a space curve without bothering with its density {line integral}.

Those who want to sit on a hyperbolic parabolic without bothering with the volume below {iterated integral}.

Those who want to climb the mountain of multi-variable functions without worrying about finding its peak {second derivative test}.

Those who want to skate inside a parabolic cylinder without worrying about how much area they covered {surface integral}.

Those who want an elegant shape without flooding it with turbulent waters {flux, Stokes’ Theorem, Divergence Theorem}.

—————————

To those who are like me. Those who want to ride across the vector space without bothers. Those who want to experience the pure beauty of surfaces and curves. **We n

I am working with an environment with a continuous action space. The agent needs to output 4 continuous values for the action space. I'm using REINFORCE, so I built a small net that, given a (31,)-sized observation space, outputs a 4x1 vector (https://www.pettingzoo.ml/sisl/multiwalker).

My loss is:

loss= -torch.sum(torch.log(prob_batch)*expected_returns_batch)

So it requires an array of action probabilities for the actions that were taken and the discounted rewards. For this reason, I recomputes the action probabilities for all the states in the trajectory and subsets the action-probabilities associated with the actions that were actually taken with the following two lines of code:

 pred_batch = model(state_batch) 

prob_batch = pred_batch.gather(dim=1,index=action_batch .long().view(-1,1)).squeeze()

However, I receive this error:

 RuntimeError: Size does not match at dimension 0 expected index [1116, 1] to be smaller than self [279, 4] apart from dimension 1 

So the problem seems to be that the agent emits four actions at each time step, but the code expects one action only (1116 is indeed four times bigger than 279). How do I fix this?

Thanks!

Idk if this is the right subreddit, but I definitely need an answer for this question from ppl knowledged in Computer Architecture.

I understand that CPUs use SuperScalar ALUs to take multiple instructions while GPUs use 100s, if not 1000s, of smaller FP32 Vector ALUs that work on a Single Instruction in parallel with the other ALUs to output multiple Data.

But my question is, what makes one SuperScalar ALU in a CPU bigger in size compared to one FP32 Vector ALU found in a GPU. Or, in other words, why does an ALU in a CPU take up more die space (transistor density) compared to an ALU in a GPU?

If this might not be the right subreddit, pls direct me to a one where this question could be answered

The exercise asks me to give two vectors that span the null space of the matrix A below:

1 3 5 0

0 1 4 -2

By augmenting it with (0, 0) I have the matrix below in echelon form without any row operation:

1 3 5 0 0

0 1 4 -2 0

Which means that:

x1 + 3x2 + 5x3 = 0

x2 + 4x3 -2x4 = 0

Hence:

x1 = -3x2 - 5x3

x2 = -4x3 + 2x4

x3 and x4 are free

So, to get to the vectors that spans the null space:

(-3x2 - 5x3, -4x3 + 2x4, x3, x4)

which is the same of:

x2 (-3, 0, 1, 0) + x3 (-5, -4, 1, 0) + x4 (0, 2, 0, 1)

The book gives me (7, -4, 1, 0) and (-6, 2, 0, 1) as possible answers, but I don't know how to get to them, but I suppose other valid answers can be found by simply using any values for x2, x3 and x4. Is this the correct approach?