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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 the chromatic polynomial of Peterson graph?
The chromatic polynomial for the Petersen (not Peterson) graph is
pi(z) = (z - 2)* (z - 1)*z*(z^7 - 12*z^6 + 67*z^5 - 230*z^4 + 529*z^3 - 814*z^2 + 775*z - 352).
How many seconds are in 35 weeks?
35 x 7days/week x 24 hrs/day x 60 minutes/hr x 60 sec/ min= 21,168,000 seconds
Direction of vector in space is specified by?
It is an integral part of the vector and so is specified by the vector.
four
1 hour = 4
2 hours = 8
3 hours = 12
Marlon is putting his stamp collection in a new album. He has 20 stamps from Canada and 90 stamps from the U.S. Each page of the album will have the same number of stamps, but stamps from Canada and the U.S. will not appear on the same page. If he puts the greatest possible number of stamps on each page, how many pages will he use? Marlon is putting his stamp collection in a new album. He has 20 stamps from Canada and 90 stamps from the U.S. Each page of the album will have the same number of stamps, but stamps from Canada and the U.S. will not appear on the same page. If he puts the greatest possible number of stamps on each page, how many pages will he use? Marlon is putting his stamp collection in a new album. He has 20 stamps from Canada and 90 stamps from the U.S. Each page of the album will have the same number of stamps, but stamps from Canada and the U.S. will not appear on the same page. If he puts the greatest possible number of stamps on each page, how many pages will he use?
What is the Solution of logarithmic equation?
The answer to the question depends on the nature of the equation. Generally speaking it will involve exponentiation (raising the log base to a power).
What is the formula for the determinant of a 3 x 3 matrix?
If the matrix is { a1 b1 c1}
{a2 b2 c2}
{a3 b3 c3}
then the determinant is
a1b2c3 + b1c2a3 + c1a2b3 - (c1b2a3 + a1c2b3 + b1a2c3)
What is the value of the expression 8 if ƒ 7?
The value of the expression 8 is always 8, whatever f is.
Addition identity.
Is 2.50 per gallon a unit rate?
Yes, a unit rate is the amount for one single unit. In this case, 'per gallon' indicates for 1 single gallon.
How many levels of algebra are there?
In terms of courses at schools, the answer depends on how the school chooses to divide the subject.
How do you graph a function with no domain?
You can't.
If f: D --> C where D is the domain of the function f and C is its codomain and D = Ø, then there are no d Є D. Therefore there are no c Є C : f(d) = c. Thus there are no ordered pairs (d, c) to graph.
What does symbolic mean in math?
In math, symbolic logic is simply expressing a mathematically logical statement through the use of symbols. For instance, one could always write down the phrase, "one plus one equals two," but using symbolic logic, that statement can be expressed much more succinctly as 1 + 1 = 2.
A better example is:
The indefinite integral of one divided by the quantity one minus the square of x with respect to x is equal to one half multiplied by the natural logarithm of the quotient of the quantities one plus x and one minus x with the constant of integration added to this result
Symbolically written, that statement is expressed as:
∫ [1/(1 - x2)] dx = ½ ln[(1 + x)/(1 - x)] + C,
which is a whole heck of a lot easier to write!
What is the formula for determinant of a 3x3 matrix?
To answer this question, let me establish an example 3 x 3 matrix named "A":
A= [a b c]
[d e f]
[g h i]
The formula I will give you, called co-factor expansion, works for any size square matrix, so you could use it to find the determinant of a 2 x 2, 3 x 3, all the way up to an n x n matrix. To find the determinant, pick any row or column in the matrix. It will make your work much easier if you choose a row or column that has many zeroes in it.
A general notation that is often used to find the determinant of a matrix is to use straight bars in place of the brackets surrounding the matrix contents. So, if I was to say mathematically that I was finding the determinant of the above example matrix, I could write it as:
det(A)= |a b c|
|d e f|
|g h i|
This notation will be used in the formula, so it is important to know this.
For the sake of an arbitrary example, let us suppose I chose Row 1 of the matrix as my chosen row. To find the determinant of this matrix, I will perform the following calculation:
(-1)2(a)|e f| + (-1)3(b)|d f| + (-1)4(c)|d e|
|h i| |g i| |g h|
This is the specific application of this general formula to the example matrix:
(-1)i+j(aij)det(A1)
In this formula, i and j are the row and column addresses, respectively, of a given matrix element. So, like in our specific application, when Row 1 was chosen as our subject row, the first term was (-1)1+1(A11)det(A1). The element "a" is in the first row, first column of the matrix, mean i=1 and j=1, therefore the superscript of (-1) is 1+1=2. A11 is simply the value held in the address i=1, j=1 of the matrix A. For this application, A11 was "a". det(A1) is the determinant of the submatrix A1. This submatrix has no formal nomenclature, I simply call it this for ease of explanation. A1 is the matrix created by "crossing out" the row and column that belong to the matrix element A11. In this application, that means it is the submatrix that is left after crossing out a, b, c, d, and g, which is simply the 2 x 2 matrix e,f;h,i. Performing this same process for the remainder of the matrix elements in Row 1 will yield the determinant of the matrix. So, the "generalized" form of the specific application above is:
(-1)1+1(A11)det(A1) + (-1)1+2(A12)det(A2) + (-1)1+3(A13)det(A3)
where A1 is the submatrix created by crossing out Row 1 and Column 1, A2 is the submatrix created by crossing out Row 1 and Column 2, and A3 is the submatrix created by crossing out Row 1 and Column 3.
A final note is how to calculate the determinants of the submatrices. For a 3 x 3 matrix, its submatrices are all 2 x 2. For 2 x 2 matrices, a simple formula exists that makes this easy:
|a b| = (ad) - (bc)
|c d|
For higher-dimension matrices, the submatrices also become larger, making the computation much more intensive.
How would you factor 24x3 plus 72x2 plus 30x?
24x^3 + 72x2 + 30x = 6x*(4x^2 + 12x + 5)=6x*(2x + 1)*(2x + 5)
It is the input value to the function called "f(x)"
If f(x)= x + 1, an input value of 2 ( x equals 2) would give us 2 + 1, or 3.
What is contradiction in algebra?
There are many ways of interpreting "contradiction" in mathematics. Some meanings are:
- Contradiction as in proof. You attempt to give the counter-proof of the theorem, but the counter-proof fails to work.
- Contradiction as in mathematical logic. If biconditional fails, we include the slash through the double arrows pointing left and right at opposite directions.
- Contradiction as in negation of the clause.
Yes, because otherwise addition and subtraction are not defined.
