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Browse College Linear Algebra

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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?

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