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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 does Multiplicative Identity Property look like?
The multiplicative property is the fact that any number multiplied by one will stay the same. i.e.
x(1)=x
Can a Hermitian Matrix possess Complex Eigenvectors?
Yes. Simple example:
a=(1 i)
(-i 1)
The eigenvalues of the Hermitean matrix a are 0 and 2 and the corresponding eigenvectors are (i -1) and (i 1).
A Hermitean matrix always has real eigenvalues, but it can have complex eigenvectors.
If a linear transformation acts on a vector and the result is only a change in the vector's magnitude, not direction, that vector is called an eigenvector of that particular linear transformation, and the magnitude that the vector is changed by is called an eigenvalue of that eigenvector.
Formulaically, this statement is expressed as Av=kv, where A is the linear transformation, vis the eigenvector, and k is the eigenvalue. Keep in mind that A is usually a matrix and k is a scalar multiple that must exist in the field of which is over the vector space in question.
How do you make 2x-5y equals 5 into points?
2x-5y=5
In order to solve this linearly(in points), you must plug in numbers into x or y, then solve the equation for the other variable.
For example: if you plug 0 into x, or into y. You will need to do this with at least two points, three two be safe.
2(0)-5y=5 2x-5(0)=5
0-5y=5 2x=5
0-5y-0=5-0 2x/2=5/2
-5y=5 x=5/2
-5y/-5=5/-5 (5/2,0)
y=-1 You then create a graph and plot the points. Next, draw a
(0,-1) connecting the two points, and...voila! you have put 2x-5y=5
into points on a line.
How do you write 35.5 billion in standard form?
The word "billion" as used in English-speaking countries means 109 (a one, followed by 9 zeros). Therefore, to convert this to standard form, you simply move the decimal point 9 positions to the right - filling out with zeros as needed.
How do you convert decimeter to decameter?
Multiply by 100.
10 decimeters = 1 meters = 1/10 decameter
Is the set of all 2x2 invertible matrices a subspace of all 2x2 matrices?
I assume since you're asking if 2x2 invertible matrices are a "subspace" that you are considering the set of all 2x2 matrices as a vector space (which it certainly is).
In order for the set of 2x2 invertible matrices to be a subspace of the set of all 2x2 matrices, it must be closed under addition and scalar multiplication.
A 2x2 matrix is invertible if and only if its determinant is nonzero. When multiplied by a scalar (let's call it c), the determinant of a 2x2 matrix will be multiplied by c^2 since the determinant is linear in each row (two rows -> two factors of c). If the determinant was nonzero to begin with c^2 times the determinant will be nonzero, so an invertible matrix multiplied by a scalar will remain invertible. Therefore the set of all 2x2 invertible matrices is closed under scalar multiplication.
However, this set is not closed under addition. Consider the matrices {[1 0], [0 1]} and {[-1 0], [0 -1]}. Both are invertible (in this case, they are both their own inverses). However, their sum is {[0 0], [0 0]}, which is not invertible because its determinant is 0.
In conclusion, the set of invertible 2x2 matrices is not a subspace of the set of all 2x2 matrices because it is not closed under addition.
How would you graph y equals 5x-12?
Select any tow values of x, say x1 and x2.Calculate y1 = 5*x1 - 12 and y2 = 5*x2 - 12.
Mark the points (x1, y1) and (x2, y2).
Join them with a straight line and extend in both directions.
What two numbers multiplies out to -600 and adds up to ten?
In the case I would first write ALL factors of 600 and find the differences.
1x600
2x300
3x200
4x150
5x120
6x100
8x75
10x60
12x50
15x40
20x30
24x25
The one that subtracts to give the answer is -20 and 30
In math, group theory is a branch of what is called "abstract algebra" that study special sets of objects called "groups".
Example: you may have studied the following facts about adding numbers.
1.) Every number has a negative of itself. (for any x there is a -x)
2.) Zero added to any number leaves that number the same. (x+0=x)
3.) No matter where you put the parentheses, addition turns out the same. For example, (x+y) +z = x+(y+z)
Therefore, numbers -- combined with the operation of addition -- form a "group" because of these three attributes.
If you learn group theory, you will discover that not just numbers obey these properties. Things like geometric symmetries, permutations, and matrices can all be described as belonging to groups.
What is the value answer of 1x10 to the power of negative 5.3?
5.0119x10 to the power of -6
or 0.0000050119
What are the basic properties of Boolean algebra?
Two first order properties are:
Atomic: ∀x x=0 ∨ ∃yy≤x ∧ atom(y)
Atomless: ∀x ¬atom(x)
They are explained at the source I linked.
Can you give an example to show that division is not associative?
Consider 75/15/3
(75/15)/3 = 5/3 = 1.66...
75/(15/3) = 75/5 = 25
Why is weathering and erosion related?
They are both the result of physical change brought about by natural forces such as wind or water.
What is the value of 1939 dime?
8-5-11>>> The 1939 Mercury Head dime is a common high mintage coin. If it's in collectible condition, values for most coins are $3.00-$5.00.
How do you figure out whether a matrix has a determinant?
Any n x n (square) matrix have a determinate. If it's not a square matrix, we don't have a determinate, or rather we don't care about the determinate since it can't be invertible.
How will volume of a balloon change if pressure remains constant but temperature increases?
The volume will increase in proportion to the increase in absolute temperature.
