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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?
 3D 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)} ... ?
 BigO 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
3dimensional 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.
 CayleyHamilton Theorem [04/19/1999]

Can you give me a proof of the CayleyHamilton 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. Twobytwo 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 greatcircle 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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