Summary of basic probability theory Math 218, …

Summary of basic probability theory Math 218, Mathematical Statistics D Joyce, Spring 2016 Sample space. A sample space consists of a un-derlying set , whose elements are called outcomes, a collection of subsets of called events, and a function Pon the set of events, called a probability

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Introduction to Modern Algebra - Clark University

Algebra became more general and more abstract in the 1800s as more algebraic structures were invented. Hamilton (1805{1865) invented quaternions (see section2.5.2) and Grassmann

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Practice Integration Z Math 120 Calculus I - Clark University

Z 3 p 7vdv. Since you can rewrite the integrand as 3 p 7v1=3, therefore its integral is 3 4 3 p 7v4=3 + C: 4. 18. Answer. Z 4 p 5t dt. The integral of 4 p 5 t 1=2 is equal to 8 p 5 t1=2 +C. You could also write that as 8 p t=5 + C. 19. Answer. Z 1 3x2 + 3 dx This integral equals 1 3 arctanx+ C. 20. Answer. Z x4 6x3 + ex p x p x dx.

Covariance and Correlation Math 217 Probability and ...

Correlation. The correlation ˆ XY of two joint variables Xand Y is a normalized version of their covariance. It’s de ned by the equation ˆ XY = Cov(X;Y) ˙ X˙ Y: Note that independent variables have 0 correla-tion as well as 0 covariance. By dividing by the product ˙ X˙ Y of the stan-dard deviations, the correlation becomes bounded ...

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Determinants, part III Math 130 Linear Algebra

2x + 3y 4z = 2 3x 2y + 5z = 7 First, compute the determinant of the 3 3 co-e cient matrix. = 1 1 3 2 3 4 3 2 5 = 54 Next, replace the rst column by the constant vec-tor, and compute that determinant. x = 6 1 3 2 3 4 7 2 5 = 27 Then in the unique solution, x = x= = 1 2. Next, replace the second column by the constant vector, and compute that ...

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Directional derivatives, steepest a ascent, tangent planes ...

normal to the surface. If x is any point in R3, then rf(a) (a x) = 0 says that the vector a x is orthogonal to rf(a), and therefore lies in the tangent plane, and so x is a point on that plane. Example 2 (Continuous, nondi erentiable func-tion). You’re familiar with functions of one vari-able that not continuous everywhere. For example,

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Some linear transformations on R2 Math 130 Linear Algebra

Example 8 (Projections). Consider the 2 2 matrix 1 0 0 0 . It sends the vector x y to the vector x 0 . This particular transformation is a projection from 2-space onto the x-axis that forgets the y-coordinate. Projections have 1-eigenspaces and 0-eigenspaces. What are they for this projection? This projection will map our leaf to the x-axis as ...

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Kernel, image, nullity, and rank Math 130 Linear Algebra

homogeneous system Ax = 0. Furthermore, these two lines are parallel, and the vector 2 4 1=2 3=2 0 3 5shifts the line through the origin to the other line. In summary, for this example, the solution set for the nonhomogeneous equation Ax = b is a line in R3 parallel to the solution space for the homogeneous equation Ax = 0.

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[1] Eigenvectors and Eigenvalues - MIT Mathematics

[4] Computing Eigenvectors [5] Computing Eigenvalues [1] Eigenvectors and Eigenvalues Example from Di erential Equations Consider the system of rst order, linear ODEs. dy 1 dt = 5y 1 + 2y 2 dy 2 dt = 2y 1 + 5y 2 We can write this using the companion matrix form: y0 1 y0 2 = 5 2 2 5 y 1 y 2 : Note that this matrix is symmetric. Using notation ...

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1 Eigenvalues and Eigenvectors - Calvin University

1 = (1,0,3) and u 2 = (1,1,3). It is a fact that all other eigenvectors associated with λ 2 = −2 are in the span of these two; that is, all others can be written as linear combinations c 1u 1 +c 2u 2 using an appropriate choices of the constants c 1 and c 2. Example: Find the eigenvalues and associated eigenvectors of the matrix A = −1 2 0 ...

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Eigenvalues and Eigenvectors §5.2 Diagonalization

0 3 1 0 0 3 1 A: Show that A is not diagonalizable. Solution: Use Theorem 5.2.2 and show that A does not have 3 linearly independent eigenvectors. I To nd the eigenvalues, we solve det( I A) = 1 1 1 0 + 3 1 0 0 + 3 = ( 1)( +3)2 = 0: So, = 1; 3 are the only eigenvalues of A: Satya Mandal, KU Eigenvalues and Eigenvectors x5.2 Diagonalization

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Eigenvalues and Eigenvectors - MIT Mathematics

P is symmetric, so its eigenvectors .1;1/ and .1; 1/ are perpendicular. The only eigenvalues of a projection matrix are 0 and 1. The eigenvectors for D 0 (which means Px D 0x/ fill up the nullspace. The eigenvectors for D 1 (which means Px D x/ fill up the column space. The nullspace is projected to zero. The column space projects onto itself.

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Eigenvalues and Eigenvectors - MIT Mathematics

P is symmetric, so its eigenvectors .1;1/ and .1; 1/ are perpendicular. The only eigenvalues of a projection matrix are 0 and 1. The eigenvectors for D 0 (which means Px D 0x/ fill up the nullspace. The eigenvectors for D 1 (which means Px D x/ fill up the column space. The nullspace is projected to zero. The column space projects onto itself.

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Eigenvalues and Eigenvectors

If all 3eigenvalues are distinct →-−%≠0 Hence, /1"=0, i.e., the eigenvectors are orthogonal (linearly independent), and consequently the matrix !is diagonalizable. Note that a diagonalizable matrix !does not guarantee 3distinct eigenvalues.

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Eigenvalues and Eigenvectors

© 2012 Pearson Education, Inc. Slide 5.1- 10 EIGENVECTORS AND EIGENVALUES ! The scalar λ is an eigenvalue of A if and only if the equation has a nontrivial solution,

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Eigenvalues, Eigenvectors, and Diagonalization

Find all of the eigenvalues and eigenvectors of A= 1 1 0 1 : The characteristic polynomial is ( 1)2, so we have a single eigenvalue = 1 with algebraic multiplicity 2. The matrix A I= 0 1 0 0 has a one-dimensional null space spanned by the vector (1;0). Thus, the geometric multiplicity of this eigenvalue is 1.

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Eigenvalues and eigenvectors of rotation matrices

λ2 − 1 = 0, (12) which yields the eigenvalues, λ = ±1. The interpretation of this result is immediate. The matrix R(θ) when operating on a vector ~v represents a reflection of that vector through a line of reflection that passes through the origin. In the case of λ = 1 we have R(θ)~v = ~v, which means that ~v is a

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