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 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. Eigenvectors and Matrices [09/01/1997] Can you please give me an example of a practical use of eigenvalues and eigenvectors? Eigenvectors of a Covariance Matrix [07/04/2004] Why do the eigenvectors of a covariance matrix point in the direction of highest variance of the initial data set? Eliminating Multiple Radicals in an Equation [02/23/2006] How can I solve equations containing different radicals, such as cuberoot(y) + fourthroot(x) = 1? Equation for Angle Formed by Two Vectors [05/31/2000] Do I need to find the vector equation of OP and OQ, or of PO and OQ, when finding the angle POQ using the formula cos(theta)=a.b/ab? Equation of a Line in Three or More Dimensions [05/18/2000] Can the equation y = mx + b be used to define a line in three dimensions? What about four or more dimensions? Expansion by Minors [10/26/1999] How do you evaluate the determinant of a 3x3 matrix using expansion by minors? Explaining the Determinant [11/16/1997] I am trying to understand what the determinant of a matrix actually is. Explaining the Dot Product [04/05/1998] Exactly what does the dot product represent? Explanation of General Solution to Difference Equations [02/08/2006] I know that the general solution to a difference equation like y(n) = Ay(n-1) + By(n-2) is given by y(n) = as^n + bs^n when y is in the form s^t. But I don't really know why that's the case. Can you explain where that comes from? Finding Area Using a Determinant [05/25/2007] Given two vectors defining a parallelogram, the area can be found using the determinant of the 2x2 matrix representing those vectors. I can't figure out why that works. Can you explain it? Finding a Transformation Matrix [12/29/2003] Vectors a1(4,2,1), a2(5,3,2), a3(3,2,1) and b1(-1,4,0), b2(4,3,1), b3 (-5,7,-3) are two bases for V3. What is the transformation matrix from first to second base? Finding Eigenvalues and Eigenvectors [06/27/2007] In a 3x3 square matrix, how would you find the eigenvalues and eigenvectors? Finding Eigenvectors from Eigenvalues [08/24/2005] Given a 3 x 3 matrix, find the eigenvalues and the corresponding eigenvectors. Finding Parametric Equations for a Line [05/24/2000] How can I find the parametric equations for the line of intersection of, and the angle between, two planes given by x-2y+z=0 and 2x+3y-2z= 0? Finding the Area of an Irregular Polygon in 3-D [02/04/2009] Given the coordinates of eight points in three-dimensional space, with no known angles, how can I find the area of the polygon they define? Finding the Jordan Canonical Form [04/29/1998] I need help in finding the Jordan Canonical Form of a matrix. Finding Two Orthogonal Vectors in R3 [12/15/2003] I'm supposed to find two non-zero vectors in R3 that are orthogonal, but I'm pretty confused about the whole idea. Can you explain it to me? Fitting Equation to Data with Regression Techniques [04/20/2007] I have a set of data and I'm trying to find a best-fit equation. I've used linear regression with little success, and I don't know much about how to do other non-linear regressions. Can you get me started? General Form of a Line [5/19/1996] What is the equation is for the general form of a line in three dimensions? in n dimensions? What would be a test for collinearity of points in three dimensions? in n dimensions? Gradient and Maximum Increase of a Function [07/19/2005] Why does the gradient point in the direction of the maximum increase of a function? Several resources make the statement, but no one explains it. Can the dot product be used to justify it? Intersecting Vectors and the Dot Product [04/24/1998] Each of the following geometrical theorems can be proved with vectors, using the dot product... Intersection of Ellipsoid and Plane [05/04/2007] I need to find the general equation of the ellipse resulting from the intersection of an ellipsoid and a plane. When I solve the system of two equations, I wind up with an xy term and need to know how to eliminate that from the final equation of the ellipse. The Intersection of Two Subspaces [10/11/1998] If w1 and w2 are subspaces of the vector space V, how can I prove that their intersection is also a subspace of V? Intersection Point of Two Lines [07/22/2003] I want to find the intersection point of two lines (in 3D) defined by a point on each line and their direction vectors V1 and V2. Inverse of a 3x3 Matrix [04/25/2001] How can I obtain the inverse of a 3x3 matrix? Inverse of a Matrix [12/05/1997] I need an algorithm to compute the inverse of a matrix. Invertible Matrices [12/07/2001] Show that if A and B are nxn invertible matrices, then A^(-1)=(A+B)^(- 1)+(A+AB^(-1)A)^(-1). Irreducible Polynomials [06/06/2001] If provided with an irreducible polynomial, how can you prove that it is indeed irreducible? E.g. the polynomial x^8+x^4+x^3+x+1 (Hex: x'11B'). Linear Algebra and Proving a Subspace [02/04/2004] (a) The set Sm = {(2a,b - a,b + a,b) : a,b are real numbers. Under the operations of matrix addition and multiplication, prove that this is a subspace of M22. (b) Find a basis for this subspace and give the dimension of the subspace. Linear Algebra - Definition of Orthonormal [10/28/1997] Show that the transition matrix P from one orthonormal basis to another is unitary, that is, P*P = I. Line of Best Fit For Points in Three Dimensional Space [11/22/2005] How can I find the line that most closely fits a set of points in 3D space, like the least squares line does in two dimensions? Matrix Inversion by the Cayley-Hamilton Theorem [02/27/1998] What is the Cayley-Hamilton Theorem is and how it is used to find the inverse of a matrix? Matrix Multiplication [09/04/1997] I am doing a project in Algebra 2 and must research matrix multiplication. Matrix Pattern Tells Solution Set? [9/4/1996] Is there a pattern for reduced row-echelon form matrices that can differentiate whether the solution set has no, one, or infinitely many solutions? Matrix Rank [09/29/2001] I want to find the rank of a matrix, but I don't know how to compute rank. Matrix Representation of Complex Numbers [11/15/2000] How can I find matrices A and B such that (A+B)^-1 = A^-1 + B^-1? Matrix Theory [06/06/2001] I would like to know how to calculate the Jordan canonical form of the matrix with columns: [(1 -4 -2 -3 -8),(0 1 -1 -1 -2),(-1 -3 0 -3 - 7),(1 2 1 4 5),(0 1 1 1 4)]. Maximum Difference, Longitude and Latitude [04/10/2001] Find the maximum longitude and latitude difference between two points on Earth 1000 kilometers apart. Minimum Matrix Multiplication [02/16/2003] Find a way to multiply n matrices using a minimum number of multiplications of the entities. Multiplicative Inverse in Finite Field GF(2^8) [02/23/2005] Is the multiplicative inverse of a 4x4 matrix in GF(2^8) the same as the inverse of any 4x4 matrix? Page: []

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