Algebra Definition and 999 Threads

I seem to have stuck with this simple algebra problem.$$ \frac {(L - 2.r)^2 } { L^2} = 1/2 $$ $$ 2( L^2 + 4r^2 - 4rL) = L^2 $$ $$ L^2 + 8 r^2 - 8r L = 0 $$ $$ (L - 2 \sqrt 2 r )^2 = 8rL - 4\sqrt 2. rL $$ $$ (L - 2 \sqrt 2 r )^2 = 4rL. (2 -\sqrt 2) $$ I don't...

Indeed, if we take a vector field which dual to the covector field formed by the gradient from a quadratic interval of an 8-dimensional space with a Euclidean metric, then the Lie algebra of linear vector fields orthogonal (in neutral metric) to this vector field is isomorphic to the...

I want to take some courses that involve heavy math, so I have been learning maths on the khan academy site: precalculus, calculus, statistics etc. But one fundamental area of maths the khan academy site doesn't have is a course on linear algebra. I really need to learn and use linear algebra in...

Hi! I want to check if i have understood concepts regarding the quotient U/V correctly or not. I have read definitions that ##V/U = \{v + U : v ∈ V\}## . U is a subspace of V. But v + U is also defined as the set ##\{v + u : u ∈ U\}##. So V/U is a set of sets is this the correct understanding...

Let V = C[x] be the vector space of all polynomials in x with complex coefficients and let ##W = \{p(x) ∈ V: p (1) = p (−1) = 0\}##. Determine a basis for V/W The solution of this problem that i found did the following: Why do they choose the basis to be {1+W, x + W} at the end? I mean since...

The options are $$rank(B)+null(B)=n$$ $$tr(ABA^{−1})=tr(B)$$ $$det(AB)=det(A)det(B)$$ I'm thinking that since it's invertible, I would focus on the determinant =/= 0. I believe the first option is out, because null (B) would be 0 which won't be helpful. The second option makes the point that...

This is the question: This is what I did. I have no other idea what else I can do to solve this confusing problem.

First, a little context. It's been a while since I last posted here. I am a chemical engineer who is currently preparing for grad school, and I've been reviewing linear algebra and multivariable calculus for the last couple of months. I have always been successful at math (at least in the...

Hello I am looking for an introductory linear algebra book. I attend university next year so I want to prepare and I want to become an engineer. I have a good background in the prerequisites, except I don't know anything about matrices or determinants. I am looking for the more application side...

I am currently enrolled in Multivariate Calculus and am looking to get build up a solid base of mathematics for undergraduate physics curriculum. I am looking for a Linear algebra book that will aid me in my quest. I currently own Axler's Linear Algebra Done Right, but I fear it is too...

Logic and equations seem to have come out of nowhere in this question. I have been unable to understand where these equations come from and why they are used. Can someone describe the logic for the steps in the question?

It says in any textbook (for example, in classical text «Theory of matrices» by P. Lankaster) on matrix theory that matrices form an algebra with the following obvious operations: 1) matrix addition; 2) multiplication by the undelying field elements; 3) matrix multiplication. Is the last one...

Summary:: Where can I find further discussion of the algebra(s) of “basic reflections” (e.g. γ^2 = -1 ), mentioned in Sec. 11.5 of Penrose’s "Road to Reality"? In Roger Penrose’s Chapter 11 of Road to Reality, titled ‘Hypercomplex Numbers’, he discusses Clifford Algebra elements being...

I am reading Sheldon Axler's book: Measure, Integration & Real Analysis ... and I am focused on Chapter 2: Measures ... I need help in order to make a meaningful start on verifying the first part of Axler, Example 28 ... The relevant text reads as follows: Can someone please help me to make...

I am reading Sheldon Axler's book: Measure, Integration & Real Analysis ... and I am focused on Chapter 2: Measures ... I need help in order to make a meaningful start on verifying the first part of Axler, Example 28 ... The relevant text reads as follows:Can someone please help me to make a...

Good Morning While I am an engineer, my math training was deficient. On my own, I made up for it, but it was a struggle. For example, I always thought Algebra was the study of rate problems (e.g.: "If a train is going one way at one speed and the other is going that way at another, where will...

I've tried to answer but I've been coming up with nonsense answers, but I'll show my method and see if its right or wrong so I can get pointed in the right direction. .35x + .45(35 - x) = 4.25, I think this is where my error is but I'm unsure. Multiply by 100 to clear the decimals and I get...

Summary:: Linear algebra 1.Let a a fixed vector of the Euclidean space E, a is a fixed real number. Is there a set of all vectors from E for which (x, a) = d the linear subspace E / 2. Let nxn be a matrix A that is not degenerate. Prove that the characteristic polynomials f (λ) of the matrix A...

I haven't gotten to multiple variables yet in algebra, but I tried to solve it this way. x - 12 = 3y -72 with y = - 12 + x Then I multiply 3x - 36 -72 = 3x -108 Next step is +108 to both sides to get 3x= 108, and then x = 36 So she was 36 72 years ago and is 96 now. The math seems to...

Suppose ##\nu## is a measure on some ##\sigma##-algebra ##\mathcal{A}##. Then we must have for all ##A \in \mathcal{A}## either ##A## or ##A^c## is finite, but not both. Because otherwise ##\nu(A)## is undefined or not well defined. I've verified that ##\lbrace \emptyset, X \rbrace## and...

Continued Fractions are themselves quite odd to learn and see in modern Algebra courses. The convergents are the numbers that we get when stop the division process at any stage. I know the laws related to convergents, let ##p_n## denote the numerator of ##n##th convergent and ##q_n## the...

∈Was wondering if anyone here could help me with an explanation as to how Axler arrived at a particular step in a proof. These are the relevant definitions listed in the book: Definition of Matrix of a Linear Map, M(T): Suppose ##T∈L(V,W)## and ##v_1,...,v_n## is a basis of V and ##w_1...

Hello, guys! I would like to know your opinion and discuss this extension of real numbers: https://mathoverflow.net/questions/115743/an-algebra-of-integrals/342651#342651 In essence, it extends real numbers with entities that correspond to divergent integrals and series. By adding the rules...

Posting for my son (who does not have an account here): He's a sophomore math major in college and is looking for a good book on Lie algebra and Lie Groups that he can study over the summer. He wants mathematical rigor, but he is thinking of grad school in theoretical physics, so he also wants...

Hello, can anyone help how i prove this logic rule? I am not sure whether i have to draw a digital circuit or something. if someone could help me solve it showing the steps they took i'd appreciate thanks

Help me solve this word problem please & explain how you did it-thanks so much, I 'm tearing my hair out. Jason purchased a can of tomato sauce and three pounds of apples for $\$7.05$. Gina purchased a can of tomato sauce and 5 pounds of apples for $\$11.35$. What is the price of the tomato...

prove that $2+8{\sqrt{-5}}$ is unit and irreducible or not in $\mathbb Z+\mathbb Z{\sqrt{-5}}$.

prove that u(z+zw)={+1,-1,+w,-w,+w^2,-w^2}

Evening, The reason for this post is because as the title suggests, I have a question concerning matrix transformation. These are essentially test prep problems and I am quite stuck to be honest. Here are the [questions](https://prnt.sc/riq7m0) and here are the...

I am just starting a QM course. I hope these are reasonable questions. I have been given my first assignment. I can answer the questions so far but I do not really understand what's going on. These questions are all about so(N) groups, Pauli matrices, Lie brackets, generators and their Lie...

Hello folks, I am currently finishing up a class on linear algebra, covering vector spaces, bases and dimension, geometry of n-dimensional space, linear transformations and systems of linear equations. I am only getting accustomed to proof writing for the first time in this course. However, I...

I do not know how to proceed.

##\int \frac{e^x (2-x^2)}{(1-x) \sqrt{1-x^2}} dx## I tried using substitution x = sin θ but still can not solve it. I guess I have to get rid the term ex but do not know how Thanks

three sub algebra of Unitary group (6) as 1. U(5) . 2. SU(3) 3. O(6) here the three chains in attachment is attached. I want to know how these chains are understands in group theory.

I have confusions about representation theory. In the following questions, I will try to express it as best as possible. For this thread say representation is given as ρ: L → GL(V) where L is the Lie group(or symmetry group for a physicist) GL(V) is the general linear...

I was solving a problem and got stuck in two aspects: 1) Geometric issue. Alright, I understand that the coordinates of the lower cylinder are $$( -R \theta_1, R)$$ The coordinates of the upper cylinder are: $$( x_1 + 2R \sin \theta, 3R - 2(R-\cos \theta))$$ I get that the ##x##...

For the following statement: V = R ≥ 1; x ⊕ y = max (x,y), with z = 1 My attempt is as follows: Should R3 be z ⊕ (x ⊕ y)? I am confused at to the notation of this rule. Moreover, I am struggling to find examples and answers of such problems in linear algebra online. Should I always view such...

Hi I have noticed that while I have the grasp of the theoretical underpinnings of linear algebra, I need work on applying it to geometric problems (think computer vision and rigid body motion). So, I am looking for a book that allows me to practice 3D geometry problems. Is there any obvious...

I want to teach myself math, and am trying to start with intermediate algebra. What are some of the best textbooks on this subject, especially for someone trying to learn this without a class?

In Quantum Mechanics, by Wigner's theorem, a symmetry can be represented either by a unitary linear or antiunitary antilinear operator on the Hilbert space of states ##\cal H##. If ##G## is then a Lie group of symmetries, for each ##T\in G## we have some ##U(T)## acting on the Hilbert space and...

Li=1/2*∈ijkJjk, Ki=J0i,where J satisfy the Lorentz commutation relation. [Li,Lj]=i/4*∈iab∈jcd(gbcJad-gacJbd-gbdJac+gadJbc) How can I obtain [Li,Lj]=i∈ijkLk from it?

My idea was to write out the formulas for the orthogonal q vectors in terms of the input vectors using the basics of gram-schmidt. Then, I would rewrite those equations suhc that the a vectors were written in terms of the q vectors. And then, try to find some matrix which would capture the...

The problem as given in the book: The price of a dress is marked down 20% on May 1. On May 25th the reduced price is marked down an additional 15% to \$51. (a) What percent of the original price is the final sale price? (b) "What is the original price? All I could think was trying to work...

in the language of general relativity,we know that we can write $$\nabla_{V}W $$ in this form such that: $$\nabla_{V}W = = w^i d ( V^j e_j)/du^i = w^j e^i (V^j e_j ) = W( V)$$ where $$w^i * d/ (du^i) =W$$ will act on the vector V where $$W = w^i d( ) /du^i $$ and W is a vector as a...

When people graduate and have their degrees in engineering or physics or mathematics or what they may have done, some of these people will use some mathematics, very often which is some-what complicated (or not) arithmetic. Why will some people choose to strictly avoid using variables in the...

I took Algebra I, II, and precalculus in high school, but I graduated high school some time ago and would like to prepare for calculus before I go back to school. My question is if a textbook that covers both Algebra and Trigonometry will sufficiently prepare me for calculus (I plan on taking...

Hello! I'm a beginner in discrete math and don't actually know how to solve the bool algebra and logc problems. Sorry for errors in formulas - it's my first post here. I have some tasks that I want someone could help for me to solve. Step-by-step solutions would be really good, to really know...

This image shows the equations. I managed to almost get equation 5, but my partial derivative is not squared but instead multiplied by mu, and also I don't have a factor of 1/2. Here is an image of the work I have. I'm sorry for any sloppiness. I tried to be as concise as possible when writing...

So this question is a math question having to do with me calculating the rate of population growth starting from a population of 100,000. I have already gotten the first 3 steps done (sex ratio, ratio of cycle time, and pregnancy ratio) after a week among those in the fertile timeframe...

I've attached a picture from the problem, the given equations, and my work so far. I knew I had a problem when My units wouldn't check out. This should just be simple algebra but It's not working. Please help me to understand what I am doing wrong.