Ask Dr. Math College Archive

Dr. Math Home || Elementary || Middle School || High School || College || Dr. Math FAQ

 TOPICS This page:   linear algebra    Search   Dr. Math See also the Internet Library:   linear algebra COLLEGE Algorithms Analysis Algebra    linear algebra    modern algebra Calculus Definitions Discrete Math Exponents Geometry    Euclidean/plane      conic sections/        circles      constructions      coordinate plane      triangles/polygons    higher-dimensional      polyhedra    non-Euclidean Imaginary/Complex   Numbers Logic/Set Theory Number Theory Physics Probability Statistics Trigonometry Browse College Linear Algebra Stars indicate particularly interesting answers or good places to begin browsing. Multiplying Sparse Matrices [12/01/2000] Is there a reasonably fast way to multiply two matrices that are very big, but sparse (they have many zeros)? Natural Numbers Coprime to 6 [02/16/1998] Let N(x) denote the number of natural numbers less than x which are coprime to 6. Show that lim as x goes to infinity of [N(x)/x] = 1/3. Nearest Point on a Great Circle [05/27/2002] Given points A, B, and C on the surface of a unit sphere, find the point P on the great circle defined by A and B that is nearest to C. One-to-one Proof [06/25/1998] Let T(x) = Ax, where A is a 2 X 2 matrix. Show that T is one-to-one if and only if the determinant of A is not zero. Orthogonal Distance Regression Line [07/05/2005] How can I calculate the standard deviation of the error in the independent and dependent variables when I want to fit a straight line with both variables subject to error? Orthogonal Distance Regression Planes [07/30/2003] I have a set of data points that I have collected from an experiment. I want to fit a 3D plane (best-fit) into these points (the points are in the form (x1,y1,z1), (x2,y2,z2),...) in order to evaluate my results. Orthogonal Matrices [01/07/1998] Prove that if A is an orthogonal matrix, the determinant of A is either 1 or -1. Pane Rotation - Computer Graphics [11/14/1996] How do I find a formula for rotating a pane in space? Parity of Permutations [7/8/1996] I can't remember how to prove that permutations are either odd or even. A Perfectly Reflecting Sphere [09/09/2000] How can I find the coordinates of each intersection of a laser beam fired from one of the "poles" of a perfect sphere into a perfectly mirrored interior of the sphere? Point of Intersection between Two Lines [07/08/2003] Given lines with coordinates (x1,y1) and (x2,y2), and (x3,y3) and (x4,y4), find whether (and where) these lines intersect between coordinates. Point within a Triangle [05/29/2003] I have the coordinates of the three corners of a equilateral triangle ABC. How can I decide whether an arbitrary point (X,Y) lies in the plane of the triangle? Population Dynamics and the Leslie Matrix [11/04/2000] What are the Leslie matrix and the age histogram vector for a population of animals? Positive Definite Matrices [08/12/2008] How can I tell if a matrix is positive definite, and if it's not, how can I make it be so? Powers of Matrices [6/27/1996] Would you send me some information/links about matrices and exponentials? Principal Axis Theorem [12/10/1997] Could you give me applications of the principal axis theorem? Product of Upper Triangular Matrices [09/23/2003] Show that the product of two upper triangular matrices is an upper triangular matrix. Proof of Circle Theorem by Vectors [05/03/2001] Prove, using the vector scalar product, that the angle in a semicircle is always 90 degrees (the hypotenuse being the diameter, and the sides meeting on the perimeter). Proof of Interesting Geometric Vector Theorem [09/17/2005] O is the center of the circumscribing circle of triangle ABC and H is its orthocenter. Prove that vector OH is equal to the sum of the vectors OA, OB and OC. Proof of Reflective Property of the Hyperbola [07/08/2004] A discussion of how to prove that a ray of light aimed at one focus of a hyperbola which hits one 'arm' will reflect off that arm and reach the other focus if we imagine that the arms function as mirrors. Proofs on Idempotent Matrices [05/09/2000] How can I determine all idempotent 2x2 matrices over R, and how can I prove that in an integral domain, the only idempotents are 0 and 1? Properties of Determinants [10/23/2000] What properties of determinants do we use to evaluate them, and how? How do we put determinants in row-echelon form? Proving Concurrence Using Vectors [10/17/2005] How do you prove that angle bisectors are concurrent using vectors? I have proved this using coordinate geometry, but I do not know how to find the point of intersection using vectors. Rotating a Plane about a Point in Space [07/03/2004] Discussion of a method of rotating the three points of a right triangle around a point to align with a given plane in a particular way. Rotating Vectors [12/14/1995] If I were a vector and I wanted to turn to face another vector how would I do it? Rotation Matrix about a Point Other Than the Origin [12/18/2006] I know that in order to rotate a point around the origin you use the rotation matrix with sine and cosine. I was wondering how you rotate a point around a point other than the origin? Rotation Matrix Using Trig Functions [12/15/2005] I know what the rotation matrix is for an angle of rotation of theta, but I don't understand why it works. Rotations in Three Dimensions [11/11/1999] How can I rotate something about an axis or about a line in 3D? Shortest Distance Between Two Vectors [04/07/1998] How do you find the shortest distance between two 3-dimensional lines? Singular Value Decomposition [11/04/2005] Can you explain what kinds of difficulties may arise while trying to find a singular value decomposition? For example, can we find eigenvectors for corresponding eigenvalues in every situation? Skew-Symmetry [02/17/1997] Using skew-symmetry to prove statements about matrices. Solve Eigenvalue of Complex Matrix [7/17/1996] Please tell me about the two methods to solve the eigenvalues of complex matrices - the Hause-Gibbson method and the Jacobi method. Solving Linear Equations in Computer Programs [02/13/2003] What is the best and simplest method in computer programs for solving linear equations: Cramers method; Gaussian Elimination method; Inverse matrix method? Some Algebra Problems [6/1/1996] If z=(3-2i)^1/2 then find z^-... Special Unitary Groups In Physics [11/09/1998] Could you explain the groups SU(2) and SU(3)? They are central to descriptions of quantum chromodynamics. Square Root of Matrix A [06/29/2003] How can we find the square root of matrix A... ? The Stationary Vector [11/13/2000] In Markov processes, what does a stationary vector tell you and how do you find it? Subspaces of R^3... [06/22/1998] Find a basis and determine the dimension of the subspace... Symmetric Matrices [10/27/1999] Find all values of a, b, and c for which a 3x3 matrix A is symmetric, where the elements of A contain algebraic expressions in a, b, and c. Tensor [10/01/1997] Is there a layman's definition for a tensor, with an example? Page: []

Search the Dr. Math Library:

 Search: entire archive just College Linear Algebra Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

[Privacy Policy] [Terms of Use]

© 1994- The Math Forum at NCTM. All rights reserved.
http://mathforum.org/