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Browse College Linear Algebra

Stars indicate particularly interesting answers or good places to begin browsing.



Equivalent Matrices [7/12/1996]
What are equivalent matrices? How do I use them?

Solving Problems Using Matrices [9/31/1995]
Given a current x,y,z coordinate (i.e. -60,-60,-60) and an angle (i.e. 90 degrees) and a plane to rotate about (i.e. z), what's the formula to solve for x,y,z?

Use of Tensors [7/23/1996]
What are the uses and properties of tensors?

3-D Coordinate Vector Transformation [8/27/1996]
What is the approach to take for transforming one coordinate system to another?

3D Geometry [11/17/1997]
You can draw a line of minimum distance between and perpendicular to two lines in 3space. I know how to get the distance and direction of this line, but I want to locate the line in 3space so that I can find its midpoint.

An Absorbing States Problem [11/28/2000]
A mouse is in one of 4 rooms. If it finds cheese in the current room, it stays there; if not, it exits to another room at random. What is the probability it will get trapped? Are there any absorbing states?

Abstract Algebra and Frobenius Automorphism [04/14/2004]
Let E be a finite field of order p^n. (a) Show that the frobenius automorphism s_p has order n. (b) Deduce from part (a) that G(E/Z_p) is cyclic of order n with generator s_p.

Adding Angles [03/27/2002]
I want to be able to add two angles without using sines or cosines, because they aren't fast enough for programming games.

Angle Between Vectors [06/07/2002]
Given vectors A and B in a plane, and vector C normal to that plane, compute the clockwise angle from vector A to vector B when viewed in the direction of the normal vector C.

Ball Bouncing off a Line Segment [05/04/2001]
If you take an arbitrary line on a 2D plane, e.g. x1y1 - x2y2, then take a point that moves about the plane, say pxpy, can you tell if this point has crossed the line at any time?

Basis, Dimension, and Orthonormality [06/19/1998]
Can you help me with questions on these topics: basis of the kernel and image, dimension of a matrix, linear dependence, and the Gram- Schmidt process?

Basis for a Vector Space in R^3 [11/25/1998]
Are the following bases for R^3: {(2,-3,1), (4,1,1), (0,-7,1)} ... ?

Big-O Notation in Matrix Multiplication [12/10/2000]
How can I prove that two n x n matrices can be multiplied in O(n^3) time? Also, is there a faster way to multiply them?

Calculating a Matrix Eigenvalue [7/16/1996]
How do you use a computer to quickly calculate the eigenvalue of a matrix with complex elements?

Calculating Angles Between Faces of a Solid [09/15/2003]
How can I compute the dihedral angles for a Great Rhombicosidodecahedron?

Calculating the Distance Between a Point and a Plane [10/12/1995]
I am trying to calculate the minimum distance between a point, located in 3-dimensional space, and a plane... I believe the minimum distance from the plane to the point is the perpendicular distance from the plane to the point. I am unsure of the equations to perform this.

Cayley-Hamilton Theorem [04/19/1999]
Can you give me a proof of the Cayley-Hamilton Theorem?

Complexity of Matrix Inversion [04/25/2001]
What is the computational complexity for the general case of inverting an NxN matrix?

Converting a Vector to a Transformation Matrix [03/19/1998]
Converting an arbitrary vector to a transformation matrix for a left- handed coordinate system.

Cramer's Rule in Action [05/08/1998]
Explaining Cramer's Rule by applying it to a system of equations.

Cross Products; Rotating in Three Dimensions [10/26/2001]
Our class understands how the cross product works, but not why or the proof behind it.

Definition of a Tensor [05/20/2001]
Can you supply a definition of a tensor similar to a vector's definition of "any quantity having magnitude and direction"?

Definition of Cartesian Tensor [12/18/1998]
What is the precise definition of a Cartesian tensor?

Deriving a 2D Rotation Matrix [11/17/2009]
How can I geometrically derive this rotation matrix?
  |x'| = |cos(theta)   -sin(theta)| |x|
  |y'|   |sin(theta)    cos(theta)| |y|


Determinant of a Matrix [11/05/1997]
Can you give us a definition for the determinant of a matrix?

Determinants [05/07/1997]
If the only numbers in a matrix are a, b, c, and d and a + d + c + d = 0, how do you show that the determinant of the matrix is zero?

Determinants of 4x4 Matrices [12/18/1996]
How do you find the determinant of a 4 x 4 matrix?

Determinants of a N x N Matrix [01/16/2004]
If S(i,j) denotes the sum of the common divisors of i and j, show that:
       |S(1,1) S(1,2) ... S(1,n)|
       |S(2,1) S(2,2) ... S(2,n)|
   Det |  .      .    .     .   |  = n!
       |  .      .      .   .   |
       |  .      .        . .   |
       |S(n,1) S(n,2) ... S(n,n)|


Determining and Checking Saddle Point for Autonomous System [06/18/2004]
Given an autonomous system defined by a pair of differential equations, I know how to find critical points and the Jacobian matrix, but how do I determine if the critical point is a saddle point?

Determining Basis for a Vector Space [11/15/1996]
What are some ways for determining whether a set of vectors forms a basis for a certain vector space?

Diagonalization of a Matrix [12/10/1998]
Diagonalize a 3x3 real matrix A (find P, D, and P^(-1) so that A = P D P^(-1)).

Diagonalizing a Matrix [04/12/2001]
Any real symmetric matrix can be diagonalized. Two-by-two integer matrices are a different story...

Distance Equations in Three Dimensions [11/26/1999]
What formula gives the distance between a point M(xo,yo,zo) and a line that has an equation in matrix form? What formula gives the distance between two lines with matrix equations?

Distance from a Point to a Great Circle [05/24/2000]
How can I find the shortest great-circle distance from point C to a great circle connecting two points A and B (specified by their latitude and longitude)?

Dual Spaces and Complementary Subspaces [08/13/2003]
What is a dual space of a vector space?

Eigensystems [11/2/1994]
I am working on alternate methods to solve polynomial equations of degree greater than 3... Could give me some insight on alternate methods of finding eigenvalues?

Eigenvalues [12/18/1998]
What is an eigenvalue and how is it used?

Eigenvalues and Hermitian Matrices [01/03/2006]
Let A be a symmetric matrix (such a matrix appears when considering multivariate distributions). Show that the eigenvalues of A are real.

Eigenvectors [05/13/1997]
What are eigenvectors?

Eigenvectors and Eigenvalues [12/22/1997]
I understand how to get the eigenvectors and eigenvalues from a 2x2 matrix, but in a 3x3 matrix I can't understand the notes!

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