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.
The following properties of big-O notation follow from the definition:
(i) if ##f(x)=O(u(x))## as ##x\rightarrow{a}##, then ##Cf(x)=O(u(x))## as ##x\rightarrow{a}## for any value of the constant ##C##.
(ii) If ##f(x)=O(u(x))## as ##x\rightarrow{a}## and ##g(x)=O(u(x))## as ##x\rightarrow{a}##...
This equation takes a present value (PV) to find mortgage payments, PMT:
Alternatively, switching V for PV and T for PMT:
V/T = r(1-r^n)/(1-r)
What is an algebraic method to solve for "r"?
Can it not be solved for? I realize I can just find out "r" by trial by error in Excel using the PMT...
Summary: What would be a good book for learning Linear Algebra by myself in my situation (which is explained in my post below)?
I did an undergraduate Linear Algebra course about 18 years ago. The textbook we used was Howard Anton’s “Elementary Linear Algebra”. The problem is that I never...
I am looking for good textbooks in physics, algebra, and trigonometry textbooks that are up to date and a good read. I heard that Feynman’s Lectures was really good. Is it still up to date enough?
Any opinions?
Hi,
For an engineer who graduated and finished typical Cal A,B,C + Linear Algebra + ODE, what book do you recommend to start reading to be a transition to advanced pure math subjects like abstract algebra and number theory?
I did deep google search & concluded that that book supposed to include...
Problem statement : Let me copy and paste the problem as it appears in the text.
Attempt : From the "Relevant Equations" given above, we can compare to see that ##a-1 = -1## and ##a^2+2=3##. These lead (after some algebra) to the three values of ##\boxed{a=0, \pm 1}##.
Issue : The book has a...
I am trying to convert the attached picture into dirac notation.
I find the LHS simple, as it is just <ψ,aφ>=<ψIaIφ>
The RHS gives me trouble as I am interpreting it as <a†ψ,φ>=<ψIa†Iφ> but if I conjugate that I get <φIaIψ>* which is not equiv to the LHS.
*Was going to type in LaTex but I...
My HS senior son is taking AP Calc this year and struggled with the first quiz, which was a review of some more difficult algebra - factoring higher degree polynomials, simplifying complicated fractions etc. Ordered him Schaum’s Int algebra for practice problems, but curious about any advice...
So this expression is apparently in Sz basis? How can you see that?
How would it look in Sy basis for example?
The solution is following. They are putting Sz as a basis, bur how do you know that Sz is the basis here?
Thanks
Dear Everybody,
I am about to teach my first course, College Algebra at my university as an instructor of record. Most of the students take this course is just for liberal arts requirement for critical thinking. I feel like I have too high expectation of my students when I should not have too...
Some textbooks I found online ( open source )
College Trigonometry 3rd Corrected Edition - STITZ ZEAGER OPEN SOURCE MATHEMATICS
Precalculus 3rd, Corrected Edition - Lakeland Community College, Lorain County Community College
A First Course in Linear Algebra - Robert A. Beezer
Cheers.
Hello,
I have been looking for textbooks for self-studying linear algebra, which seems to be quite an important course. I have read that in order to study quantum mechanics well, one must have a very good command of linear algebra.
Some textbooks in my country are quite bad and only teach...
Summary: Hello! I'm an high school student and i want to study more math but I'm not sure where to start. Should i first study linear algebra or calculus?
Hello! I'm an high school student and i want to study more math but I'm not sure where to start. Should i first study linear algebra or...
Problem Statement : I copy and paste the problem from the text to the right.
Attempt (mine) : Given the inequality ##\dfrac{x}{x+2}\le \dfrac{1}{|x|}##. We see immediately that ##x\ne 0, -2##. At the same time, since ##|x|\ge 0\Rightarrow \frac{x}{x+2}\ge 0##.
Now if ##\frac{x}{x+2}\le...
(I could solve the problem but could not make sense of the solution given in the text. Let me put my own solutions below first).
1. Problem Statement : I copy and paste the problem to the right as it appears in the text.
2. My attempt : There are three "regions" where ##x## can lie.
(1)...
Summary: Calculating a "fixed profit" on inventory
[Mentor Note -- after an initial move to the schoolwork forums, this thread turns out to be a General Math question after all (see posts #5-6), so it is moved back to the General Math forum]
I have 1000 products for sell in a store.
Some make...
Problem statement : Let me copy and paste the problem as it appears in the text on the right.Attempt (myself) : By looking at ##\large{\sqrt{x+2}\ge x}##, from my Relevant Equations above, we have the following :
1. Outcome ##\mathbf{x \ge 0}##, since square roots are always positive.
2...
In [this post][1] user William Ryman asked what would happen if we try to build "complex numbers" with shapes other than circle or hyperbola in the role of a "unit circle".
[Here][2] I proposed three shapes that could work. The common principle behind them being
that if the unit curve is...
Hi all,
I have a formula which i might have expected to have brackets, but it hasn't, so i need to correctly interpret it.
Could anyone please offer any pointers as to how it should read?
The formula is '1.2 . h - 0.2 . x'
Can i assume that it should be '(1.2 . h) - (0.2 . x)'?
Thanks
Zamb
*Kindly note that i created this question (owned by me).
My Approach,
##\dfrac {(x+y)(4x+6y)}{(5x-5y)}##=##-1##
##(x+y)(4x+6y)=-5x+5y##
##\dfrac {4x+6y}{-5x+5y}##=##\dfrac {1}{x+y}##
to get the simultaneous equation,
##4x+6y=1##
##-5x+5y=x+y##
...
##4x+6y=1##
##-6x+4y=0##
giving us...
The first image is the question and the second is the answer.
My thinking is let's say North is positive, and South is negative. Fixed point O is the starting point. Then the question becomes +(2a-b)-(3a+2b). The answer should be -a-3b. I cannot fathom why the book gives the answer as a+b. Any...
I'm self-studying tensors from a book that doesn't have exercises. The book is Semi-Riemannian Geometry by Newman. To get a better feel for index manipulation, tensor results and calculations, I'm looking for a book that has many exercises in these topics.
I'd be grateful if those knowledgeable...
Hello everyone, I would like to get some help with the above problem on signals and linear projections. Is my approach reasonable? If it is incorrect, please help. Thanks!
My approach is that s3(t) ad s4(t) are both linear combinations of s1(t) and s2(t), so we need an orthonormal basis for the...
My intuition about the Lie algebra is that it tries to capture how infinitestimal group generators fails to commute. This means ##[a, a] = 0## makes sense naturally. However the Jacobi identity ##[a,[b,c]]+[b,[c,a]]+[c,[a,b]] = 0## makes less sense. After some search, I found this article...
I think the answer is an even function as the function ##x^2## is an even function and thus, is symmetrical w.r.t. Y axis. The question I have is how to do this problem algebraically. I tried to graph some functions on GeoGebra to verify my answer.
a) ##y = ln(x^2)##
b) ##y = sin(x^2)##...
Computer languages handle the scope of variables in a precise way so that if one symbol, such as "k" is used in different contexts, the program keeps these separate. When sophisticated human beings re-use symbols in writing mathematics, they can keep things straight, but I don't think they...
Is Advanced Linear and Matrix Algebra by Nathaniel Johnston a good book on linear algebra? Will it teach me all I need to know? Is there any calculus in it despite the name? I never took a course on linear algebra so I'm looking for something that teaches everything and includes calculus with...
Problem statement : Let me copy and paste the problem statement from the text :
Attempt at solution : I could not solve the problem reducing the L.H.S into the R.H.S. However, I could solve the problem by expanding the R.H.S. into the L.H.S., though it is less than satisfactory. Below is my...
Problem : Let me copy and paste the problem statement as it appears in the text, as shown above.
Attempt : I can sense there is an "elegant" way of doing this, but I don't know how. I show below my attempt using ##\text{Autodesk Sketchbook}##. I hope am not violating anything.
Ok so I have...
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
$$\frac {1} {2700} + \frac {1} {3930n^2} + 10^{-5}$$
$$\frac {3930n^2+2700+2700*3930n^2*10^{-5}}...
Problem statement : Let me copy and paste the problem as it appears in the text :
Attempt : I am afraid this looks like a very difficult problem, despite being at the elementary level (high school). My glance through the text shows that the authors have gone about reducing the first set of...
Problem: Given the line L: x = (-3, 1) + t(1,-2) find all x on L that lie 2 units from (-3, 1).
I know the answer is (3 ± 2 / √5, -1 ± 4/√5) but I don't know where to start. I found that if t=2, x= (-5, 5) and the normal vector is (2, 1) but I am not sure if this information is useful or how...
Hey! :giggle:
Give for the following expressions of relation algebra the equivalent expression in relational calculus.
1. $\sigma_{B=A}(R(A,B,C))$
2. $\pi_{B,C}(R(A,B,C))$
3. $R(A,B,C)\cup S(A,B,C)$
4. $R(A,B,C)\cap S(A,B,C)$
5. $R(A,B,C)\setminus S(A,B,C)$
6. $R(A,B,C)\times S(D,C,E)$
7...
First of all, I attached pictures of the very last algebra textbook that I have finished studying. I'm going the self taught route. I really loved this book because it had lots of examples, practice exercises, quizzes and even tests! It also had answers in the back. It's currently my favorite...
I’m reading Weinberg’s QFT books, and stacking how to solve problem 15.4.
Weinberg says there is no simple lie algebra with just four generators, but I have no idea how to approach this problem. If the number of generators are only one or two, it can easy to say there is not such a simple lie...
Until I understand how to use maple for my steps by steps algebra manipulation feature (which I learned it has), I'll use PF for some help in the algebra.
I want to derive the expression for ##D=-\Delta \sin^2 \theta## on page 313 in Equation (11.89).
Attachments of printscreen below.
I wrote...
First thing to notice is that ##L## and ##L \circ L## are precisely equal linear maps.
What we know
$$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...
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)...