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Abstract Algebra
Have you ever wondered what would happen if you redefined some of the basic rules of algebra so that concepts you took for granted, like AB = BA, no longer apply? Abstract algebra does just that through the study of the properties that define algebraic structures. Post all questions about fields, rings, group theory, vector spaces, and the axioms that define them into this category.
1,849 Questions
What is an idempotent give examples of idempotent matrix.?
An idempotent is a matrix whose square is itself. Specifically, A^{2}=A. For example the 2x2 matrix
A= 1 1
0 0
is idempotent.
When did algebra get its name?
Algebra comes from the Arabic word al-jabr, which appeared in the title of a book Hidab al-jabr wal muqubala (roughly translated as The book of restoration and cancellation".
This was written in Baghdad in the early 9th Century by the mathematician Mohammed ibn-Musa al-Khowarizmi.
Is an invertible idempotent matrix the identity matrix?
The assertion is true.
Let A be an idempotent matrix. Then we have A.A=A. Since A is invertible, multiplying A-1 to both sides of the equality, we get A = I.
Q. E. D
Prove that a matrix a is singular if and only if it has a zero eigenvalue?
Recall that if a matrix is singular, it's determinant is zero. Let our nxn matrix be called A and let k stand for the eigenvalue. To find eigenvalues we solve the equation det(A-kI)=0for k, where I is the nxn identity matrix.
(<==)
Assume that k=0 is an eigenvalue. Notice that if we plug zero into this equation for k, we just get det(A)=0. This means the matrix is singluar.
(==>)
Assume that det(A)=0. Then as stated above we need to find solutions of the equation det(A-kI)=0. Notice that k=0 is a solution since det(A-(0)I) = det(A) which we already know is zero. Thus zero is an eigenvalue.
What is an unknown change problem?
In Unknown Change problems, the middle number is the unknown number.
Example 1 - Unknown Change (minus)
162 apples were in an apple tree. A gust of wind blew the tree, and some apples fell to the ground. 124 apples remained. How many apples fell from the tree?
(Answer = 38)
Example 2 - Unknown Change (plus)
Rochester, NY received 1 foot of snow in 24 hours. Over the next few days, it continued snowing. When the skies finally cleared, there was a total accumulation of 5 feet, 6 inches of snow. How much snow fell after the first day?
(Answer = 4 feet, 6 inches)
To divide something into a ratio like this you need to first see how many parts are involved.
In this instance 3 to 5 totals 8.
So we need to divide the number by 8, then multiply by 3 and by 5
728 / 8 = 91
91 x 3 = 273
91 x 5 = 455
273 + 455 = 728
so, 273 : 455 = 3 : 5
What is the primeter of a rectangle 8m x 3m?
Since the formula for the perimeter of a rectangle is 2 times the lenght plus 2 times the width (P = 2l + 2w). P = 2(8) + 2(3) P = 16 + 6 P = 22m
In mathematics, sets are simply collections of objects. Set theory is the branch of mathematics that studies these collections of objects.
For more information, please refer to the related link below.
What is a median in a set of numbers?
If its a odd set of numbers then the median will be (n+1/2)th term.
where,
n=set of numbers like
2,4,5 then the median will be (3+1/2)th term=2nd term=4.
therefore the median is 4
And if its a even set of numbers like
1,4,7,9,6,8 then the median will be the (sum of mid numbers/2)
7+9/2=8
therefore the median is 8
What is a linear transformation?
linear transformation can be define as the vector of 1 function present in other vector are known as linear transformation.
Boolean algebra is a division of mathematics that deals with operations on logical values and incorporates binary variables.
What is an example of a Tarski monster group?
I am curious as to why someone would pose a question about an advanced topic like this on this site, when a quick search for "Tarski monster group" brought up this:
http://en.wikipedia.org/wiki/Tarski_monster_group
To chase up the reference there you need access to a good college or university library.
How do you prove a Ring to be commutative?
To prove a ring is commutative, one must show that for any two elements of the ring their product does not depend on the order in which you multiply them. For example, if p and q are any two elements of your ring then p*q must equal q*p in order for the ring to be commutative.
Note that not every ring is commutative, in some rings p*q does not equal q*p for arbitrary q and p (for example, the ring of 2x2 matrices).
If two rows of a determinant are interchanged what is true of the resulting determinant?
The resulting determinate is the negative, or opposite, of the original determinant.
There are 10 numbers in all including 0. The first space can be filled in 9 ways (as we have to exclude 0). The second through sixth spaces can be filled in 10 ways as 0 can be included.
Totally, 9 x 10 x 10 x 10 x 10 x 10 or 9 x 10^5 digits can be formed if repetition is allowed.
Why is the word algebra in boolean algebra?
You have Boolean operators (such as AND & OR) on variables, rather than mathematical operations (+ - etc). The variables can only have one of two states (values) though (True/False, on/off, 1/0).
What is the difference between integrated algebra and algebra 1?
Algebra 1 is a class/course that is on a higher level than Algebra.
It is a matrix indicating the out of balance transaction figures between two entities who are subsidiaries of one another or subsidiaries of some other entity. The matrix is periodicaly reveiwed to diagnose and clear the out of balance figures. The out of balance figures can also be due to foreign exchange transactions between two entities whereby gain/loss arise on revaluation of balances periodically.
Abstract algebra is a field of mathematics that studies groups, fields and rings, which all belong to algebraic structures. Algebraic structure and abstract algebra are actually close to each other due to their similarity in topics.
