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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
Is there any connection between Evariste Galois's group theory and symmetry?
There sure is, and a major connection at that.
Consider a finite set of n elements. The symmetric group of this set is said to have a degree of n. The symmetric group of degree n (Sn) is the Galois group of the general polynomial of degree n. In order for there to be a formula involving radicals that solve the general polynomial of degree n, such as the quadratic equation when n = 2, that polynomial's corresponding Galois group must be solvable. S5 is not a solvable group. Therefore, the Galois group of the general polynomial of degree 5 is not solvable. Thus the general polynomial of degree 5 has no general formula to solve it using radicals.
This was huge result, and one of the first real applications, for group theory, since that problem had stumped mathematicians for centuries.
What are the Idiosyncrasies of matrix algebra?
idiosyncrasies of matrix are the differences between matrix algebra and scalar one. i'll give a few examples.
1- in algebra AB=BA which sometimes doesn't hold in calculation of matrix.
2- if AB=0, scalar algebra says, either A, B or both A and B are equal to zero. this also doesn't hold in matrix algebra sometimes.
3- CD=CE taking that c isn't equal to 0, then D and # must be equal in scalar algebra. Matrix again tend to deviate from this identity.
its to be noted that these deviations from scalar algebra arise due to calculations involving singular matrices.
What are rational numbers between negative 1 and positive 1?
There are an infinite number of rational numbers between -1 and +1.
Take for example: N / (N+1) for any positive integer N: 1/2, 2/3, 3/4, 4/5, 5/6, etc.
This is just between +1/2 and +1, and still doesn't cover all of them. Consider 1/N for any non-zero integer: 1/1, 1/2, 1/3, 1/4, 1/5, etc. 1/(-1)=-1, 1/(-2)=-1/2, -1/3,-1/4, etc. There are an infinite number of schemes that you can come up with where the magnitude (absolute value) of the denominator is greater than the magnitude of the numerator.
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!
Let x = pounds of $0.75 candy, and let y = pounds of $1.25 candy.
From the question, we know the total pounds must equal 9, and the total price should be 9 pounds at $0.96 per pound. Thus the total price should be 9*0.96 = $8.64.
We can form two equations; one for total pounds, and one for total price:
x+y = 9
0.75x+1.25y = 8.64
Solve for x in the first equation: x = 9-y
Plug that into the second equation for x: 0.75(9-y) +1.25y = 8.64
Simplify: 6.75 - 0.75y + 1.25y = 8.64
More: 6.75 + 0.5y = 8.64
More: 0.5y = 1.89
Divide both sides by 0.5: y = 3.78
Plug that into the first equation: x + 3.78 = 9
Solve for x: x = 5.22
Thus, we need 5.22 pounds of $0.75 candy and 3.78 pounds of $1.25 candy.
What is three times a number a equals 14.4?
To find a, all you have to do is solve 3 X a = 14.4 for a. To do that, simply divide both sides of the equation by 3, which gives you a = 4.8.
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.
What is the difference between scalar multiplication and matrix multiplication?
Matrix multiplication is when you multiply the numbers inside different matricies.[topleft#1]Xtopleft#2=top left
topright#1XBottomleft=top right
bottom left X Topleft=top left
bottom rightX bottom right=bottom right
Scalar multiplication A number out side a matrix multiplies all parts of the matrix
y = x² - 2x - 8
Swap x and y to find the inverse:
x = y² - 2y - 8
Now get y by itself:
y² - 2y = x + 8
(Complete the square):
y² - 2y + 1 = x + 9
(y - 1)² = x + 9
y - 1 = ±√(x + 9)
y = 1 ±√(x + 9)
Then there are two functions:
f(x) = 1 - √(x + 9)
and
f(x) = 1 + √(x + 9)
but an inverse can only be one function.
The functions above are a mirror image of the original function, but it is not an inverse (graph these functions).
There is no inverse.
Everything above is completely wrong. This is the inverse: (x+8)/2
I don't know where the x² came from in the first answer, but to work out the question as asked:
if y=2x-8
y+8 = 2x
(y+8)/2 = x
so, as the above answer stated, (x+8)/2 is the answer.
Can a component of a vector be greater than the vector itself?
No, the magnitude of a vector (in Euclidean space) is the square root of the sum of the squares of its components. This value can never be greater than the value of one of its own components.
v = √((vx)2 + (vy)2 + (vz)2)
v2 = (vx)2 + (vy)2 + (vz)2
(vx)2 = -(vy)2 - (vz)2 + v2
vx = √(-(vy)2 - (vz)2 + v2)
Can vx > v?
Substituting:
√(-(vy)2 - (vz)2 + v2) > √((vx)2 + (vy)2 + (vz)2).
Simplified:
v2 > (vx)2 + 2(vy)2 + 2(vz)2.
Substituting again:
(vx)2 + (vy)2 + (vz)2 > (vx)2 + 2(vy)2 + 2(vz)2.
Simplifying again:
0 > (vy)2 + (vz)2.
This results in a fallacy, since 0 can't be greater than a positive number. This wouldn't work even if both vy and vz were 0.
What is the correct classification of x2 plus 3x plus 8?
It is a quadratic polynomial with integer coefficients..
Addition identity.
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.
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.
Why only square matrix have determinant?
The square matrix have determinant because they have equal numbers of rows and columns.
<<>> Determinants are not defined for non-square matrices because there are no applications of non-square matrices that require determinants to be used.
How do you decide what operation you use in order of operations?
You memorize the rules that are considered standard.
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).
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)
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?
