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Browse High School Linear Algebra
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    Vector questions.



Matrix Algebra [06/01/1998]
Solving for X in the matrix product A*X - X*B = C.

Matrix Division? [06/26/2006]
Is it possible to divide matrices?

Matrix Encoder and Decoder [07/22/2003]
How does the decoder work?

Matrix Inverses [04/07/1998]
Can you find a 2x2 matrix A such that A^2 = I? Can you find a 2x2 matrix that has no inverse?

Matrix Multiplication [09/04/1997]
I am doing a project in Algebra 2 and must research matrix multiplication.

Matrix Multiplication [12/18/1998]
Why does matrix multiplication work? Why are the rows multiplied and added with the columns?

Matrix Multiplication [10/11/2003]
I'm trying to multiply two matrices. I know what the dimensions of the product have to be, but how do I do the actual multiplication?

Matrix Questions: Addition, Inverse [8/6/1996]
(1 2) + (-1 -2) =? and if A = (3 4), does A inverse exist?

Matrix Rank [09/29/2001]
I want to find the rank of a matrix, but I don't know how to compute rank.

Matrix: Why Use m and n? [09/06/2001]
Why do we use m and n, with x between them, to represent order in a matrix?

Matrix with No Inverse [06/13/2005]
Are there are 3x3 matrices that do not have inverses? If so, can you give me an example?

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.

More on the Intersection of Two Lines in Three Dimensions [02/17/2009]
How to isolate for "a" in the vector equation a(V1 x V2) = (P2 - P1) x V2, with a suggestion for computer coding.

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?

Multiplying a Matrix By a Scalar [08/27/1998]
When you multiply a scaler times a matrix, the result is a new matrix...

Multiplying Matrices and Identifying Dimensionality [07/16/2003]
I understand adding and subtracting matrices, but not multiplying them. Also, how can I find the dimensionality of the following matrix...?

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.

Pane Rotation - Computer Graphics [11/14/1996]
How do I find a formula for rotating a pane in space?

Parallelepipeds [05/06/1997]
What is the formula for finding the total surface area of a parallelepiped?

Parametrics [12/18/1996]
You're in 3-D space at point A, you want to get to point B, and you know the coordinates to point B from point C (but B is moving). What heading do you need to set in order to meet point B?

Point Inside or Outside Triangle? [08/02/2001]
I have coordinates of three vertices of the triangle and coordinates of the point.

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?

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 About Matrices [06/02/1999]
How do I prove that in two 2 x 2 matrices, A and B, AB - BA will never be the multiplication identity matrix?

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?

Prove Choice of Scalars Unique [03/13/2001]
Suppose B = {v1,v2, ... vn} is a basis for a vector space V. Let u element of V be any vector. Prove that the expression of u in terms of the basis B is unique.

Raytrace of a Star Sapphire [12/11/1996]
How do I find the rotations necessary to create a raytrace of a star sapphire so that the star always faces the camera?

Rearranging Matrices [11/5/1995]
My problem involves matrices, specifically 9x9 size. I would like to know how to determine what order to rearrange the rows so the numbers in the diagonal add up to the maximum possible value.

Redundant Sets of Equations [02/07/1997]
Is there a solution to the set consisting of these three equations: x + y + z = 50, 2x + 3y + 4z = 158, and 4x + 3y + 2z = 142?

A Re-Framing Inversion: Transforming the Coordinates in Between [03/27/2012]
Given some function that pass through (0, 0) and (1, 1), a student struggles to invert it after zooming in on two in-between points and re-centering on them as the new origin and (1, 1). Doctor Peterson obliges, starting with a sketch and simpler labels.

Rotación de Véctores (3-D Vector Rotation) [6/19/1996]
Hay alguna fórmula para rotar véctores en el espacio sobre los tres ejes de coordenadas?

Rotating a cube in space [12/5/1994]
I am a high school teacher, with a student interested in writing a program to show a cube on the screen and then rotate it by moving the corners around. He has asked me for equations. I don't have any. Do you?

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.

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?

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