Algebra Definition and 999 Threads

Use your counters to do each of the following multiplication problems using the definition of multiplying a whole number by an integer. Use the following example as a model. Example Multiply: $2\times -6\implies 2\times -6= RRRRRR + RRRRRR = RRRRRRRRRRRR=-12$ why are they using 6 Rs...

So my final equation is: ##\frac {1} {2700} + \frac {1} {3930n^2} + 10^{-5}## I need to boil this down, the learning materials has the following working, but I can't seem to get it $$\frac {1} {2700} + \frac {1} {3930n^2} + 10^{-5}$$ $$\frac {3930n^2+2700+2700*3930n^2*10^{-5}}...

Or is reading a proofs book enough

Problem statement : Let me copy and paste the problem as it appears in the text : Attempt : I am afraid this looks like a very difficult problem, despite being at the elementary level (high school). My glance through the text shows that the authors have gone about reducing the first set of...

Problem: Given the line L: x = (-3, 1) + t(1,-2) find all x on L that lie 2 units from (-3, 1). I know the answer is (3 ± 2 / √5, -1 ± 4/√5) but I don't know where to start. I found that if t=2, x= (-5, 5) and the normal vector is (2, 1) but I am not sure if this information is useful or how...

Hey! :giggle: Give for the following expressions of relation algebra the equivalent expression in relational calculus. 1. $\sigma_{B=A}(R(A,B,C))$ 2. $\pi_{B,C}(R(A,B,C))$ 3. $R(A,B,C)\cup S(A,B,C)$ 4. $R(A,B,C)\cap S(A,B,C)$ 5. $R(A,B,C)\setminus S(A,B,C)$ 6. $R(A,B,C)\times S(D,C,E)$ 7...

First of all, I attached pictures of the very last algebra textbook that I have finished studying. I'm going the self taught route. I really loved this book because it had lots of examples, practice exercises, quizzes and even tests! It also had answers in the back. It's currently my favorite...

I’m reading Weinberg’s QFT books, and stacking how to solve problem 15.4. Weinberg says there is no simple lie algebra with just four generators, but I have no idea how to approach this problem. If the number of generators are only one or two, it can easy to say there is not such a simple lie...

Until I understand how to use maple for my steps by steps algebra manipulation feature (which I learned it has), I'll use PF for some help in the algebra. I want to derive the expression for ##D=-\Delta \sin^2 \theta## on page 313 in Equation (11.89). Attachments of printscreen below. I wrote...

First thing to notice is that ##L## and ##L \circ L## are precisely equal linear maps. What we know $$L \ \text{is injective} \iff \ker(L)=\{0\}$$ $$\ker L' = \{ x \in \Im(L) \ | \ L'(x)=0\}$$ $$\Im(L)=\{ x \in V \ | \ \exists \ v \in V \ \text{such that} \ L(v)=x\}$$ Besides, we notice...

please offer me examples of: a) 3 vector spaces over the same field; and b) the same vector space over 3 fields.

I have the following problem in multilinear algebra: Let ##W## and ##V## be real finite-dimensional vector spaces, ##V^*## is the dual space of ##V## Let ##L:W \times V \rightarrow \mathbb{R}## be a non-degenerate bilinear map Define ##g:W \rightarrow V^*## by ##g(w)(v)=L(w,v)## To prove: ##g##...

We only worry about finite vector spaces here. I have been taught that a subspace ##W## of a vector space ##V## has a complementary subspace ##U## if ##V = U \oplus W##. Besides, I understand that, given a finite vectorspace ##(\Bbb R, V, +)##, any subspace ##U## of ##V## has a complementary...

Show that ##U = span \{ (1, 2, 3), (-1, 2, 9)\}## and ##W = \{ (x, y, z) \in \Bbb R^3 | z-3y +3x = 0\}## are equal. I have the following strategy in mind: determine the dimension of subspaces ##U## and ##W## separately and then make use of the fact ##dim U = dim W \iff U=W##. For ##U## I would...

Hello all, I have a problem related to LU Factorization with my work following it. Would anyone be willing to provide feedback on if my work is a correct approach/answer and help if it needs more work? Thanks in advance. Problem: Work:

I'm looking for literature recommendations regarding tensor networks. I never came across singular value decomposition or spectral decomposition in my linear algebra classes, so I need to brush up on the relevant mathematical background as well.

Hi guys! :) I was solving some linear algebra true/false (i.e. prove the statement or provide a counterexample) questions and got stuck in the following a) There is no ##A \in \Bbb R^{3 \times 3}## such that ##A^2 = -\Bbb I_3## (typo corrected) I think this one is true, as there is no squared...

Summary:: Hi guys, i can't seem to get the correct answer. I'm wondering where did I do wrong. Can someone help me to solve this? I think I need the correct formula to prove the answer :( Given a root to 𝑥² + 𝑝𝑥 + 𝑞 = 0 is twice the multiple of another. Show that 2𝑝² = 9𝑞. The roots for 𝑥² +...

Now, i am extremelly confused about all this thing. More preciselly, i can't understand how 1.29 was obtained. It was used the A representation? How do he uses it? There is something to do with the canonical basis?

Let's say I want to study subalgebras of the indefinite orthogonal algebra ##\mathfrak{o}(m,n)## (corresponding to the group ##O(m,n)##, with ##m## and ##n## being some positive integers), and am told that it can be decomposed into the direct sum $$\mathfrak{o}(m,n) = \mathfrak{o}(m-x,n-x)...

Hello I have problems with this exercise Prove that the algebra generated by the set $S = \{ 1,x^2 \}$ is dense in $C [0, 1]$. It is $S$ dense in $C [-1; 1]$ I am thinking to apply Stone-weierstrass theorem but I don't know how to use it properly. Thanks

https://en.wikipedia.org/wiki/Klein%27s_Encyclopedia_of_Mathematical_Sciences Originals are in German or French, the Japanese version cut all the historical content :( Do you think that some day we will see this published in English? Size is big, 20k pages, but it cannot be more interesting I...

What are the suitable books in linear algebra for third course for self-study after reading Linear Algebra done right by Axler and Algebra by Artin?

My guess is that since there are no rows in a form of [0000b], the system is consistent (the system has a solution). As the first column is all 0s, x1 would be a free variable. Because the system with free variable have infinite solution, the solution is not unique. In this way, the matrix is...

What are best second course(undergraduate) books in linear algebra for self-study?I have already read Introduction to Linear Algebra by Lang.

In Miles Reid's book on commutative algebra, he says that, given a ring of functions on a space X, the space X can be recovered from the maximal or prime ideals of that ring. How does this work?

Let ##\mathfrak{A}:=\operatorname{span}\left\{D_n:=x^n\dfrac{d}{dx}\, : \,n\in \mathbb{Z}\right\}## and ##\mathfrak{B}:=\operatorname{span}\left\{E_n:=x^n\dfrac{d}{dx}\, : \,n\in \mathbb{N}_0\right\}## with the usual commutation rule. My question is: How can we prove or disprove the Lie algebra...

I found this interesting video from Presh Talwalkar: Problem Statement. If: $$x + y + z = 1$$$$x^2 + y^2 + z^2 = 2$$$$x^3 + y^3 + z^3 = 3$$ Then, find the value of the higher powers such as $$x^5 + y^5 + z^5$$ The solution posted there uses the full Girard-Newton Identities. Here is an...

Using differential expressions for the generator, verify the commutator expression for ##[J_{\mu\nu},P_{\rho}]=i(\eta_{\mu\rho}P_{\nu}-\eta_{\nu\rho}P_{\mu})## in Poincare group Generator of translation: ##P_{\rho}=-i\partial_{\rho}## Generator of rotation...

So the reason why I'm struggling with both of the problems is because I find vector spaces and subspaces hard to understand. I have read a lot, but I'm still confussed about these tasks. 1. So for problem 1, I can first tell you what I know about subspaces. I understand that a subspace is a...

This seems like a simple problem, but I am a little confused by a few things. For one, what is the use of the piece of information that when they charged $100 per person they got 3000 people to come? Also, how should I proceed with the information "for every $2 decrease in price they would have...

Summary:: What pre-requisites are required in order to learn the textbook "Linear Algebra (2nd Edition) 2nd Edition by Kenneth M Hoffman (Author), Ray Kunze (Author)" Sorry if this is the wrong section to ask what the title and subject state. I read some of chapter 1 already, and that all...

Textbook answer: "If P1P2 = P2P1 then S is contained in T or T is contained in S." My query: If P1 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix}and P2 =\begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix} as far as I...

I am reading a book of Fundamental Energy Systems. The author describes the rate of change in head for a turbomachine as: $$ \frac{1}{2}[(V_1^2-V_2^2)+(U_1^2-U_2^2)+(V_{R2}^2-V_{R1}^2)] = H =U_1V_{u1} - U_2V_{u2} $$ and the static effect as: $$SE =(U_1^2-U_2^2)+(V_{R2}^2-V_{R1}^2) $$ However...

Why does Δ(BA) = AΔ(B) when A is constant? Is there a proof for this algebra? Thanks

%%% Assume that ##X/Y## is defined. Since ##\dim Y = \dim X##, it follows that ##\dim {X/Y}=0## and that ##X/Y=\{0\}##. Suppose that ##Y## is a proper subspace of ##X##. Then there is an ##x\in X## such that ##x\notin Y##. Let us consider the equivalence class: ##\{x\}_Y=\{x_0\in...

I want to show that ##[C, a_{r}] = 0##. This means that: $$ Ca_{r} - a_{r}C = \sum_{i,j} g_{ij}a_{i}a_{j}a_{r} - a_{r}\sum_{i,j} g_{ij}a_{i}a_{j} = 0$$ I don't understand what manipulating I can do here. I have tried to rewrite ##g_{ij}## in terms of the structure...

Hi PF community, recently i learned about Calculus in one variables and several, so now i'd like to study linear algebra by myself in a undergraduate level, in order to do that i need some textbooks recommendations. I'll be waiting for your recommendations :).

In the below figure how triangle OAP is right angled. I have imagined everything but I cannot imagine angle A as right angled. Thanks!

Hello everyone, I am new here, so please let me know if I am doing something wrong regarding the formatting or the way I am asking for help. I did not really know how to start off, so first I tried to just write out all the ##\mu \nu \rho \sigma## combinations for which ##\epsilon \neq 0## and...

I see that the first four equations are definitions. The problem is about the dimensions of the quotient. Why does the set Kx forms a six dimensional Lie algebra?

I'm following the lecture notes by https://www.thphys.uni-heidelberg.de/~weigand/QFT2-14/SkriptQFT2.pdf. On page 169, section 6.2 he is briefly touching on the non-abelian gauge symmetry in the SM. The fundamental representation makes sense to me. For example, for ##SU(3)##, we define the...

Can i call "x" an unknown object that varies ?

Below is the problem and the correct answer for this algebra problem is 7√2. But I cannot get to the correct answer.

b) c and d): In c) I say that ##L_h## is only self adjoint if the imaginary part of h is 0, is this correct? e) Here I could only come up with eigenvalues when h is some constant say C, then C is an eigenvalue. But I' can't find two.Otherwise does b-d above look correct? Thanks in advance!

The sum of three numbers is 95. The second number is 5 more than the first. The third number is 3 times the second. What are the numbers?

Hey! 😊 I want to prove by using the rules of boolean algebra that the following statement is always true $$\{b\land [\neg a\Rightarrow \neg b]\} \Rightarrow a$$ Since we have to use the rules of boolean algebra, we cannot use the truth table, right? Could you give me a hint how we could...

Hello, I am a very experienced Mathematician with a BSc Honours degree in Mathematics and one year MSc studies in Operational Research in Sussex and London Universities respectively. I am interested in Advanced Calculus, Algebras, Positivity in Algebraic Geometry, The standard Model, and many...

##\frac {7}{2x+2}=\frac {4x-3}{-2x-2}## ##-7(2x+2)=(4x-3)(2x+2)## ##x^2+2x+1=0## ##x=1## or ##x=-1## can we also have; ##-7=4x-3## can the ##2x+2## cancel out? i am a bit mixed up on this very simple problem...and why am i getting false on my ti nsipre...

##A^{x'} = T(A^{x})##, where T is a linear transformation, in such way maybe i could express the transformation as a changing of basis from x to x' matrix: ##A^{x} = T_{mn}(A^{x'})##, in such conditions, i could say det ##T_{mn} \neq 0##. But how to deal with, for example, ##(x,y) -> (e^x,e^y)## ?