Algebra (from Arabic: الجبر, romanized: al-jabr, lit. 'reunion of broken parts, bonesetting') is one of the broad areas of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. It includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians.
Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values. For example, in
x
+
2
=
5
{\displaystyle x+2=5}
the letter
x
{\displaystyle x}
is an unknown, but applying additive inverses can reveal its value:
x
=
3
{\displaystyle x=3}
. Algebra gives methods for writing formulas and solving equations that are much clearer and easier than the older method of writing everything out in words.
The word algebra is also used in certain specialized ways. A special kind of mathematical object in abstract algebra is called an "algebra", and the word is used, for example, in the phrases linear algebra and algebraic topology.
A mathematician who does research in algebra is called an algebraist.
Homework Statement
Rocket is accelerating constantly. Let S' be instantaneous rest frame of rocket and S be frame in which rocket is observed moving at velocity v.
Homework Equations
Given: $$ dv = dv' (1 - v^2) $$
Must prove:
$$ \frac{dv}{dt} = \frac{dv'}{dt'} (1 -...
Homework Statement
This isn't a homework problem; it's just something I'm working on and I'm a little confused as to how to go about dealing with what I have. I have several traces of Dirac's gamma matrices, and I know that the trace of an odd number of gamma matrices is zero. So my first...
Homework Statement
Given an nxn matrix, if a b exists so Ax=b has no solutions, can A be one-to-one?
Homework Equations
I understand that as a linear transformation, you need things such as (to be one-to-one as a linear trans)
1. n pivots
2. Only the trivial solution exists to Ax=0
Ax=b...
Hey,
I need help with part D2. My explanation is not right so I honestly do not know what I am suppose to write. My assignment is attached to this thread.
Homework Statement
Suppose X is a set with n elements. Prove that Bij(X) ≅ S_n.
Homework Equations
S_n = Symmetric set
≅ = isomorphism
Definition: Let G and G2 be groups. G and G2 are called Isomorphic if there exists a bijection ϑ:G->G2 such that for all x,y∈G, ϑ(xy) = ϑ(x)ϑ(y) where the...
Hi, I'm looking for a good math book for a 10 year old.
He's actually very gifted, as he's able to pick up things like factors and equalities pretty fast. The thing is I'm not a math teacher and I think he needs to go over basic geometry and arithmetic before moving on to algebra, basically...
Homework Statement
For the system of springs
a) Assemble the stiffness matrix K and the force-displacement relations, K*u = f
b) Find the L*D*L^T factorization of K. Use Matlab to solve
c) Use the boundary conditions and applied forces to find the displacements
Homework EquationsThe Attempt...
Homework Statement
I am given the follow graph and asked to find the left null space
Homework EquationsThe Attempt at a Solution
First I start by transpose A because I know that the left null space is the null space of the incidence matrix transposed. I then reduce it to reduce row echelon...
Homework Statement
No official problem, just a study guide fill-in-the-blanks with an extended simplification blank. Basically, no values were given, and it is a standard block on a standard slope with a north-east applied force pushing down on the block (not parallel to horizontal or vertical...
I'm confused about the notation
T:R^n \implies R^m
specifically about m. From my understanding if n=2 then (x1, x2). Are we transforming n=2 to another value m for example (x1, x2, x3)?
I am trying to build up a kind of mind map of the following:
Module (eg. vector space)
Ring (eg Field)
Linear algebra (concerning vectors and vector spaces, from what I understood)
Multilinear Algebra (analogously concerning tensors and multi-linear maps)
Linear maps & Multilinear maps
The...
C \in \mathbb{R}^{m \times n}, X \in \mathbb{R}^{m \times n}, W \in \mathbb{R}^{m \times k}, H \in \mathbb{R}^{n \times k}, S \in \mathbb{R}^{m \times m}, P \in \mathbb{R}^{n \times n}
##{S}## and ##{P}## are similarity matrices (symmetric).
##\lambda##, ##\alpha## and ##\beta## are...
I'm trying to create a square using algebra tiles. The question is x^2 + 4x + 5. I know how to do it without the algebra tiles but I don't know how to do it with the algebra tiles.
Can anyone give me a hand with this?
Homework Statement
Hey, I posted another question yesterday, and thanks to the kindness and brilliance of hall of ivy, I was able to solve it. However when I apply the same logic to this new question I cannot seem to get it, can someone explain or show me how to do this question.
Consider the...
Homework Statement
Consider the linear transformation T from
V = P2
to
W = P2
given by
T(a0 + a1t + a2t2) = (−4a0 + 2a1 + 3a2) + (2a0 + 3a1 + 3a2)t + (−2a0 + 4a1 + 3a2)t^2
Let E = (e1, e2, e3) be the ordered basis in P2 given by
e1(t) = 1, e2(t) = t, e3(t) = t^2
Find the coordinate matrix...
Can someone tell me if Axler's texts use proofs? I'm looking for a book that teaches proofs at the high school level. Something other than a geometry text.
Homework Statement
Prove that \dim L(\mathbb F)+\dim Ker L=\dim(\mathbb F+Ker L) for every subspace \mathbb{F} and every linear transformation L of a vector space V of a finite dimension.
Homework Equations
-Fundamental subspaces
-Vector spaces
The Attempt at a Solution
Theorem: [/B]If...
Homework Statement
I have these two equations from this paper: https://www.scribd.com/doc/299960566/Spiral Homework Equations
What are the vertical bars next to the drawn red starts supposed to indicate in this context?
I'm trying to implement these equations into a program but I'm totally...
Homework Statement
If gcd(f(x),g(x)) = 1 and m,n ∈ ℕ, show that gcd(f(x)^m, g(x)^n) = 1.
Homework EquationsThe Attempt at a Solution
So I had previously proved this for non-polynomials:
gcd(a,b)=1
then gcd(a^n,b^n)=1
Proof: a = p1*p2*...*pn
b = p1*p2*...*pm
then
a^n = p1^n*p2^n*...*pn^n...
Homework Statement
Suppose a field F has n elements and F=(a_1,a_2,...,a_n). Show that the polynomial w(x)=(x-a_1)(x-a_2)...(x-a_n)+1_F has no roots in F, where 1_f denotes the multiplicative identity in F.
Homework EquationsThe Attempt at a Solution
Strategy: We have this polynomial...
Homework Statement
Determine whether the set spans ℜ3. If the set does not span ℜ3 give a geometric description of the subspace it does span.
s = {(1, 0, 3), (2, 0, -1), (4, 0, 5), (2, 0, 6)}
Homework EquationsThe Attempt at a Solution
I am having trouble with the second part of this problem...
Homework Statement
1. Let g(x) = x^4+46.
a) Factor g(x) completely in ℚ[x].
b) Factor g(x) completely in ℝ[x].
c) Factor g(x) completely in ℂ[x].
2. Completely factor the given polynomial in ℤ_5.
[4]_5 x^3 + [2]_5 x^2 + x + [3]_5
Homework Equations
ℚ = {m/n / m and n belong to Z, m is not...
Hi All,
When operating on tables T1, T2 and obtaining a table T3, one obtains rows {##r_{3k}##} in T3 that are obtained from
rows {## r_{1i}##} and {##r_{2j} ##} by what I call "concatenation" . I wonder if there is a formal name for it.
Specifically, say, during an inner join between T1...
I'm looking for an excellent introductory linear algebra textbook for my second year pure mathematics course. My lecturer highly recommended Introduction to Linear Algebra by Marcus and Minc. She said she has searched for it for many years without success, as it is out of print. I love classic...
Hi all!
I have an important decision to make for the summer of 2016 and I need some advice from some who have taken these courses. I need one biology lab elective to graduate, but it is a field lab and it runs from from 5/13 - 6/19. Because it is a field lab, I will not be able to take other...
Dear All
Am trying to study supersymmetry algebra. We start by constructing the graded poincare algebra by considering the direct sum of the ordinary Poincare algebra L0 with a subspace L1 spanned by the spinor generators Qa where a runs from 1 to 4 ( L1 is not a lie algebra ). So now we have...
Hi PF!
I have a twin brother in the military who is trying to take physics with algebra online (for military purposes he is unable to commute to a University). I thought this course would be easy to find since nearly every school offers calculus online. I haven't been able to find any schools...
Homework Statement
Show that all square matrix (A whatever) can be written as the sum of a symmetric matrix and a anti symmetric matrix.
Homework Equations
I think this relation might be relevant : $$
A=\frac{1}{2}*(A+A^{T})+\frac{1}{2}*(A-A^{T})
$$
The Attempt at a Solution
I know that we...
Homework Statement
Let g(x) ∈ ℤ[x] have degree at least 2, and let p be a prime number such that:
(i) the leading coefficient of g(x) is not divisible by p.
(ii) every other coefficient of g(x) is divisible by p.
(iii) the constant term of g(x) is not divisible by p^2.
a) Show that if a ∈ ℤ...
Homework Statement
Find all real numbers k such that x^2+kx+k is reducible in ℝ[x].
Homework EquationsThe Attempt at a Solution
This seems like it is simple, but it is new to me so I am looking for confirmation.
We know we can find the roots of a polynomial with b^2-4ab. We want b^2-4ab to be...
Homework Statement
Let T:R-> S be a homomorphism of rings. Show that T(0_r) = 0_s.
Homework EquationsThe Attempt at a Solution
First off, the terminology used is kinda confusing. I take 0_r to be the zero in R. Is this correct? For some reason I recall my teacher quickly saying that it was...
Homework Statement
Check if L(p)(x)=(1+4x)p(x)+(x-x^2)p'(x)-(x^2+x^3)p''(x) is a linear transformation on \mathbb{R_2}[x]. If L(p)(x) is a linear transformation, find it's matrix in standard basis and check if L(p)(x) is invertible. If L(p)(x) is invertible, find the function rule of it's...
I'm learning geometric algebra. There is a very simple statement which I think is wrong. But it must be right, because all the experts say so. Arrg!
The only properties used are
1a = a1 = a
aa = 1
if b<>a then ab = -baTheir claim is that abcdabcd = -1.
Let's see:
aa = 1
abab = -abba =...
Homework Statement
Let U is the set of all commuting matrices with matrix A= \begin{bmatrix}
2 & 0 & 1 \\
0 & 1 & 1 \\
3 & 0 & 4 \\
\end{bmatrix}. Prove that U is the subspace of \mathbb{M_{3\times 3}} (space of matrices 3\times 3). Check if it contains span\{I,A,A^2,...\}. Find the...
Hello,
I'm looking for an algebra based physics book with challenging problems. The reasons is because I'm preparing for an exam (in two weeks time) which will have algebra based physics. Here are the topics:
1. Kinematics
2. Dynamics (Newton's laws, Momentum , Work & Energy)
3. Gravitation
4...
Hey everyone,
I am currently taking College Physics I which covers basic mechanics, Newtonian physics, and some other various topics. It is algebra based, as I didn't have physics in high school. I was wondering if there are any good YouTube channels that you all find helpful? I normally watch...
Homework Statement
Let R be a ring and suppose r ∈R such that r^2 = 0. Show that (1+r) has a multiplicative inverse in R.
Homework Equations
A multiplicative inverse if (1+r)*x = 1 where x is some element in R.
The Attempt at a Solution
We know we have to use two facts.
1. Multiplicative...
Homework Statement
Suppose R is a commutative ring with only a finite number of elements and no zero divisors. Show that R is a field.
Homework Equations
Unit is an element in R which has a multiplicative inverse. If s∈R with r*s = 1.
A zero divisor is an element r∈R such that there exists...
Homework Statement
I can't understand this paper. I understand the whole incidence matrix stuff, but I don't quiet get how it relates to the linear algebra. I don't know if this is allowed to do, but I will ask you questions line by line, so basically you will read the paper with me explaining...
I'm taking an abstract algebra course that uses Hungerford's "An Introduction to Abstract Algebra" 3rd Ed. And while I feel like I'm following the material sufficiently and can do most of the proofs it's hard to learn and practice the material without a solutions guide. How am I supposed to know...
Homework Statement
Find the span of U=\{2,\cos x,\sin x:x\in\mathbb{R}\} (U is the subset of a space of real functions) and V=\{(a,b,b,...,b),(b,a,b,...,b),...,(b,b,b,...,a): a,b\in \mathbb{R},V\subset \mathbb{R^n},n\in\mathbb{N}\}
Homework Equations
- Span
-Subset
The Attempt at a Solution...
Homework Statement
Let and are two basis of subspaces and http://www.sosmath.com/CBB/latexrender/pictures/69691c7bdcc3ce6d5d8a1361f22d04ac.png. Find one basis of http://www.sosmath.com/CBB/latexrender/pictures/38d4e8e4669e784ae19bf38762e06045.png and...
Hi everybody,
Let G a four dimmensionnal Lie group with g as lie algebra. Let T1 ... T4 the four generator. I would like to find à lorentzian scalar product (1-3 Signature) on it and left invariant. A classical algebra take tr (AB^t) as scalar product but I don't find à lorentzian équivalent...
I need a book that has a lot of problems and can teach me how to solve word problems well and comes with a lot of problems to solve at the end of the chapter or unit.
I want a good linear algebra textbook in order to learn to use linear algebra in physics but also to use it in more theoretical mathematics courses.
I hope that with this poll i will also help others that want to study from a proper Linear Algebra textbook.
I am currently doing aops introduction to algebra book and I can't figure out how to solve joint proportion or some ppl call it joint variation. I don't which values should i substitute for the formula, z=kxy. I don't which value should i substitute into z to the word problems. I can get through...