0
Proofs
Proof means sufficient evidence to establish the truth of something. It is obtained from deductive reasoning, rather than from empirical arguments. Proof must show that a statement is true in all cases, without a single exception.
1,294 Questions
AB equals 2 What is the value of AC?
AB=2 so A=B/2
AC=CB/2. I guess that wasn't what you were looking for though. But that all that can be done with the information.
It's a set of nodes, together with edges that have directions associated with them.
Prove that the trace of a matrix A is equal to the sum of its eigenvalues?
Given a matrix A=([a,b],[c,d]), the trace of A is a+d, and the det of A is ad-bc.
By using the characteristic equation, and representing the eigenvalues with x, we have the equation
x2-(a+d)x+(ad-bc)=0
Which, using the formula for quadratic equations, gives us the eigenvalues as,
x1=[(a+d)+√((a+d)2-4(ad-bc))]/2
x2=[(a+d)-√((a+d)2-4(ad-bc))]/2
now by adding the two eigenvalues together we get:
x1+x2=(a+d)/2+[√((a+d)2-4(ad-bc))]/2+ (a+d)/2-[√((a+d)2-4(ad-bc))]/2
The square roots cancel each other out being the same value with opposite signs, leaving us with:
x1+x2=(a+d)/2+(a+d)/2
x1+x2= 2(a+d)/2
x1+x2=(a+d)
x1+x2=trace(A)
Q.E.D.
General proofThe above answer only works for 2x2 matrices. I'm going to answer it for nxn matrices. (Not mxn; the question only makes sense when the matrix is square.)The proof uses the following ingredients:
(1) Every nxn matrix is conjugate to an upper-triangular matrix
(2) If A is upper-triangular, then tr(A) is the sum of the eigenvalues of A
(3) If A and B are conjugate, then tr(A) = tr(B)
(4) If A and B are conjugate, then A and B have the same characteristic polynomial (and hence the same sum-of-eigenvalues)
If these are all true, then we can do the following: Given a matrix A, find an upper-triangular matrix U conjugate to A; then (letting s(A) denote the sum of the eigenvalues of A) s(A) = s(U) = tr(U) = tr(A).
Now to prove (1), (2), (3) and (4):
(1) This is an inductive process. First you prove that your matrix is conjugate to one with a 0 in the bottom-left corner. Then you prove that this, in turn, is conjugate to one with 0s at the bottom-left and the one above it. And so on. Eventually you get a matrix with no nonzero entries below the leading diagonal, i.e. an upper-triangular matrix.
(2) Suppose A is upper-triangular, with elements a1, a2, ... , an along the leading diagonal. Let f(t) be the characteristic polynomial of A. So f(t) = det(tI-A). Note that tI-A is also upper-triangular. Therefore its determinant is simply the product of the elements in its leading diagonal. So f(t) = det(tI-A) = (t-a1) * ... * (t-an). And its eigenvalues are a1, ... , an. So the sum of the eigenvalues is a1 + ... + an, which is the sum of the diagonal elements in A.
(3) This is best proved using Summation Convention. Summation convention is a strange but rather useful trick. Basically, the calculations I've written below aren't true as they're written. For each expression, you need to sum over all possible values of the subscripts. For example, where it says b_ii, it really means b11 + b22 + ... + bnn. Where it says bil deltali, it means (b11 delta11 + ... + b1n deltan1) + ... + (bn1 delta1n + ... + bnn deltann). Oh, and deltakj=1 if k=j, and 0 otherwise.
Suppose B = PAP-1. Let's say the element in row j and column k of A is ajk. Similarly, say the (i,j) element of P is pij, the (k,l) element of P-1 is p*kl, and the (i,l) element of B is bil. Then:
bil = pij ajk p*kl
And the trace of B is given by:
tr(B) = bii
= bil deltali
= p*kl deltali pij ajk
= p*kl plj ajk
= deltakj ajk (since p and p* are inverses)
= ajj
= tr(A)
(4) Again, suppose B = PAP-1. Then, for any scalar t, we have tI-B = P(tI-A)P-1. Hence det(tI-B) = det(P).det(tI-A).det(P-1). Since det(P).det(P-1)=det(PP-1)=det(I)=1, we have det(tI-B) = det(tI-A).
Which postulate or theorem can be used to determine the measure of RT?
For this type of question the diagram that is with the question is needed to provide you an answer.
How do you solve a problem with perpendicular bisector?
The Answers community requires more information for this question. Please edit your question to include more context. The answer depends on what the problem is!
The line goes through the midpoint, which is halfway between points. The distances are equal to each other, and proves that they are equidistant.
Data collection typically involves several stages:
- Planning: Define the objectives, identify the data needed, and select appropriate methods for collection.
- Design: Develop a detailed plan that includes tools and instruments for collecting data, such as surveys, interviews, or observations.
- Execution: Implement the data collection process, ensuring that data is gathered systematically and ethically.
- Review: Analyze and validate the collected data to ensure accuracy and reliability before proceeding with further analysis.
What are the four parts of induction?
1.Show it works for the loest value (substitute into separate ways of finding value)
2.Assume it works for x=k
3.Prove it works for x=k+1 by rearranging into required form
4.conclude
How can you show that a set is a subset of another?
If you want to show that A is a subset of B, you need to show that every element of A belongs to B. In other words, show that every object of A is also an object of B.
How many different number combinations are in the world?
Since a number can have infinitely many digits, there are infinitely many possible combinations.
Proof to find the area of a square?
The unit of area "one square meter" or "one square foot" is DEFINED as the area of a square with sides of length 1 meter or 1 foot. This works for any unit of distance measurement. So we start with this definition. It follows that a square with sides of length n when n is an integer has area n2 square units because it can be divided into n*n= n2 small squares one unit on a side. For the area of a square with sides of fractional length, we can use a proof that calls upon similar polygons. This proves the area exists, it does NOT prove it is unique. To prove that, assume it is not uniqe and arrive at a contradiction.
We definitely need to see the drawing that goes along with that question
before we can even begin to try and answer it.
What are the examples of equal set?
Two sets are equal if they contain the same identical elements. If two sets have only the same number of elements, then the two sets are One-to-One correspondence. Equal sets are One-to-One correspondence but correspondence sets are not always equal sets.
Ex: A: (1, 2, 3, 4)
B: (h, t, m, k)
C: (4, 1, 3, 2)
A and C are Equal sets and 1-1 correspondence sets.
