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Linear Algebra
Linear algebra is the detailed study of vector spaces. With applications in such disparate fields as sociology, economics, computer programming, chemistry, and physics, including its essential role in mathematically describing quantum mechanics and the theory of relativity, linear algebra has become one of the most essential mathematical disciplines for the modern world. Please direct all questions regarding matrices, determinants, eigenvalues, eigenvectors, and linear transformations into this category.
2,176 Questions
What are the differences of linear and non-linear equation?
Linear equations have a variable only to the first degree(something to the power of 1).
For example: 2x + 1 = 5 , 4y - 95 = 3y are linear equations.
Non-linear equation have a variable that has a second degree or greater.
For example: x2 + 3 = 19, 3x3 - 10 = 14 are non-linear equations.
Which measure of cetral tendency would be appropriate to describe this data set?
None. The data set has no elements and so there cannot be any central tendency.
A system of linear equations that has soluton is?
A system of linear equations that has at least one solution is called consistent.
What is non-examples of homeostasis?
Homeostasis is defined as any self-regulating process by which biological systems tend to maintain stability while adjusting to conditions that are optimal for survival.
An example of being out of homeostasis would be to have diabetes. That's when your body is NOT making enough insulin. That would be out of homeostasis because your body's not making everything "just right". People with diabetes have to test their blood to see what their sugar level is, and self-administer shots of insulin to balance their levels.
linear mime is when you draw somwthing the use it. e.g. i draw a tv then switch it on linear mime is when you draw somwthing the use it. e.g. i draw a tv then switch it on
Why is an electric field strength a vector quantity?
for a vector quantity it must have both magnitude and direction and since it has both magnitude and direction it is therefore considered a vector
Well, if you take the rule of PEMDAS (Parenthesese, Exponent, Multiplication, Division, Addition, Subtraction) and apply it to the equation you should see this:
10-3x2
= 10-6 (because 3 times 2 is 6)
=4
What is the definition of the math pie?
Mathamatical pi is an irrational number. It classifys as an irrational number because the numbers never repeat and pi is a never ending number.
What are the three forms of a linear equation?
1. Slope-intercept form (most commonly used in graphing)
y=mx+b
m=slope
b=y-intercept
2. Standard form
ax+by=c
3. Point slope form (most commonly used for finding linear equations)
y-y1=m(x-x1)
m=slope
one point on the graph must be (x1,y1)
The coefficients and constant in one of the equations are a multiple of the corresponding coefficients and constant in the other equation.
What are the x and y values for y4x-2 and yx 13?
Unfortunately, limitations of the browser used by Answers.com means that we cannot see most symbols. It is therefore impossible to give a proper answer to your question. Please resubmit your question spelling out the symbols as "plus", "minus", "times", "equals", "squared", "cubed" etc. Please use "brackets" (or parentheses) because it is impossible to work out whether x plus y squared is x + y2 of (x + y) 2.
What does square brackets mean in chemistry?
Isn't this the wrong section? >_>
Square brackets generally refer to the concentration of whichever element or compound. For example [A] would refer the the concentration of A (generally calculated in moles/Liter or molarity).
Which is lesser cube root of 6 or fourth root of eight?
Cube root of 6 is lesser than fourth root of eight.
cube root of 6 = 1.817fourth root of 8 = 1.862
Dot product of two vectors is equal to cross product what will be angle between them?
(A1) The dot product of two vectors is a scalar and the cross product is a vector? ================================== (A2) The cross product of two vectors, A and B, would be [a*b*sin(alpha)]C, where a = |A|; b = |B|; c = |C|; and C is vector that is orthogonal to A and B and oriented according to the right-hand rule (see the related link). The dot product of the two vectors, A and B, would be [a*b*cos(alpha)]. For [a*b*sin(alpha)]C to equal to [a*b*cos(alpha)], we have to have a trivial solution -- alpha = 0 and either a or b be zero, so that both expressions are zeroes but equal. ================================== Of course one is the number zero( scalar), and one is the zero vector. It is a small difference but worth mentioning. That is is to say if a or b is the zero vector, then a dot b must equal zero as a scalar. And similarly the cross product of any vector and the zero vector is the zero vector. (A3) The magnitude of the dot product is equal to the magnitude of the cross product when the angle between the vectors is 45 degrees.
What is 'non-coincident peak demand' in an electrical system?
Non-Coincident Peak (NCP) is the individual or actual peak demands of each load in an electrical system oftentimes occurring at different hours of the day. It does not necessarily fall during system peak. (This is what i understood about NCP...I gladly welcome corrections)
Each one of the crosses represents one person that has died there, usually from a traffic accident. In the county east of where I live, a white cross is painted in the middle of the traffic lane nearest the fatality. My county uses the wooden white crosses as well.
What are linear equation used for?
Linear equations are used to find the value of missing numbers. They are also used to find the relation among the missing numbers.
For example=
x+3=4
x=4-3
x=1
The direction of the electric field vector is defined as?
Direction of the electric field vector is the direction of the force experienced by a charged particle in an external electric field.
The question implies that there is a list of statements of which one or more are true. In such circumstances would it be too much to expect that you make sure that there is something that is following?
What are the application of matrices in field of engineering?
Matrices have a wider application in engineering. Many problems can be transformed in to simultaneous equation and their solution can easily be find with the help of matrices.
What is the determinant of a 2x3 matrix?
The determinant function is only defined for an nxn (i.e. square) matrix. So by definition of the determinant it would not exist for a 2x3 matrix.
State and prove Lagrange's theorem?
THEOREM:
The order of a subgroup H of group G divides the order of G.
First we need to define the order of a group or subgroup
Definition:
If G is a finite group (or subgroup) then the order of G is the number of elements of G.
Lagrange's Theorem simply states that the number of elements in any subgroup of a finite group must divide evenly into the number of elements in the group. Note that the {A, B} subgroup of the Atayun-HOOT! group has 2 elements while the Atayun-HOOT! group has 4 members. Also we can recall that the subgroups of S3, the permutation group on 3 objects, that we found cosets of in the previous chapter had either 2 or 3 elements -- 2 and 3 divide evenly into 6.
A consequence of Lagrange's Theorem would be, for example, that a group with 45 elements couldn't have a subgroup of 8 elements since 8 does not divide 45. It could have subgroups with 3, 5, 9, or 15 elements since these numbers are all divisors of 45.
Now that we know what Lagrange's Theorem says let's prove it. We'll prove it by extablishing that the cosets of a subgroup are
- disjoint -- different cosets have no member in common, and
- each have the same number of members as the subgroup.
Lemma:
If H is a finite subgroup of a group G and H contains n elements then any right coset of H contains n elements.
Proof:
For any element x of G, Hx = {h • x | h is in H} defines a right coset of H. By the cancellation law each h in H will give a different product when multiplied on the left onto x. Thus each element of H will create a corresponding unique element of Hx. Thus Hx will have the same number of elements as H.
Lemma:
Two right cosets of a subgroup H of a group G are either identical or disjoint.
Proof:
Suppose Hx and Hy have an element in common. Then for some elements h1 and h2 of H
h1 • x = h2 • y
This implies that x = h1-1 • h2 • y. Since H is closed this means there is some element h3 (which equals h1-1 • h2) of H such that x = h3 • y. This means that every element of Hx can be written as an element of Hyby the correspondence
h • x = (h • h3) • y
for every h in H. We have shown that if Hx and Hy have a single element in common then every element of Hx is in Hy. By a symmetrical argument it follows that every element of Hy is in Hx and therefore the "two" cosets must be the same coset.
Since every element g of G is in some coset (namely it's in Hg since e, the identity element is in H) the elements of G can be distributed among H and its right cosets without duplication. If k is the number of right cosets and n is the number of elements in each coset then |G| = kn.
Alternate Proof:
In the last chapter we showed that a • b-1 being an element of H was equivalent to a and b being in the same right coset of H. We can use this Idea establish Lagrange's Theorem.
Define a relation on G with a ~ b if and only if a • b-1 is in H. Lemma: The relation a ~ b is an equivalence relation.
Proof:
We need to establish the three properties of an equivalence relation -- reflexive, symmetrical and transitive.
(1) Reflexive:
Since a • a-1 = e and e is in H it follows that for any a in G
a ~ a
(2) Symmetrical:
If a ~ b then a • b-1 is in H. Then the inverse of a • b-1 is in H. But the inverse of a • b-1 is b • a-1 so
b ~ a
(3) Transitive:
If a ~ b and b ~ c then both a • b-1 and b • c-1 are in H. Therefore their product (a • b-1) • (b • c-1) is in H. But the product is simply a • c-1. Thus
a ~ c
And we have shown that the relation is an equivalence relation.
It remains to show that the (disjoint) equivalence classes each have as many elements as H.
Lemma:
The number of elements in each equivalence class is the same as the number of elements in H.
Proof:
For any a in G the elements of the equivalence class containing a are exactly the solutions of the equation
a • x-1 = h
Where h is any element of H. By the cancellation law each member h of H will give a different solution. Thus the equivalence classes have the same number of elements as H.
One of the imediate results of Lagrange's Theorem is that a group with a prime number of members has no nontrivial subgroups. (why?)
Definition:
if H is a subgroup of G then the number of left cosets of H is called the index of H in G and is symbolized by (G:H). From our development of Lagrange's theorem we know that
|G| = |H| (G:H)
Determinant of matrix in java?
/*This function will return the determinant of any two dimensional matrix. For this particular function a two dimensional double matrix needs to be passed as arguments - Avishek Ghosh*/
public double determinant(double[][] mat) {
double result = 0;
if(mat.length 2) {
result = mat[0][0] * mat[1][1] - mat[0][1] * mat[1][0];
return result;
}
for(int i = 0; i < mat[0].length; i++) {
double temp[][] = new double[mat.length - 1][mat[0].length - 1];
for(int j = 1; j < mat.length; j++) {
System.arraycopy(mat[j], 0, temp[j-1], 0, i);
System.arraycopy(mat[j], i+1, temp[j-1], i, mat[0].length-i-1);
}
result += mat[0][i] * Math.pow(-1, i) * determinant(temp);
}
return result;
}
What are the applications of cross product and dot product?
Cross product tests for parallelism and Dot product tests for perpendicularity.
Cross and Dot products are used in applications involving angles between vectors.
For example given two vectors A and B;
The parallel product is AxB= |AB|sin(AB).
If AXB=|AB|sin(AB)=0 then Angle (AB) is an even multiple of 90 degrees. This is considered a parallel condition. Cross product tests for parallelism.
The perpendicular product is A.B= -|AB|cos(AB)
If A.B = -|AB|cos(AB) = 0 then Angle (AB) is an odd multiple of 90 degrees. This is considered a perpendicular condition. Dot product tests for perpendicular.
