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Browse College Linear Algebra

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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?

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