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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.



Definition of a Vector [02/28/2002]
I would like a "proper" definition of a vector and how concepts of "magnitude" and "direction" arise in the discussion of vectors.

Questions about Matrices [2/24/1995]
What is the inverse of a matrix, and how do you find it? How do you multiply two matrices together?

What is a Vector? [01/04/2002]
I am having trouble understanding exactly what a vector is and cannot seem to find a simple, straightforward explanation.

What is Linear Algebra? [09/06/1997]
I have absolutely no idea about linear algebra...

3D Figures and Intersections [03/04/1999]
Determining whether a line and plane intersect, and where, using vectors.

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.

3-D Points to a 2-D computer screen [07/25/1997]
I am trying to write a 3-D graphics engine that takes 8 points with their own (X,Y,Z) coordinates to plot a CUBE in 3-D space...

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?

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.

The Adjoint of a Matrix [10/21/1998]
Given the adjoint of matrix A, how would you determine matrix A?

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

Calculating Matrix Determinants by Expansion of Minors [11/19/2003]
When I calculate the determinant by expansion of minors in a 3 x 3 matrix, I always find that the result for the second row is the opposite of the result for rows 1 and 3, and I don't understand why. Can you explain it?

Coin Tosses, Dealing Cards... [12/08/1998]
Several questions on discrete math - probability and combination; deducing recurrence relations.

Coordinates of Endpoint of a Segment [11/01/2002]
Determine the coordinates of the other endpoint of a segment with an endpoint at (1,-1) and a midpoint at (2,5).

Cramer's Rule and a System of Two Equations [12/27/2003]
I have a system of two equations and two variables, 2x - 6y = 12 and -4x + 3y = 7. How do I use Cramer's Rule to solve the system?

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

Crossed up by Scalars and Vectors [02/14/2012]
A student wonders why a dot product results in a scalar, whereas a cross product produces a vector. Doctor Jerry introduces example vectors, their angles, and their perpendiculars to illustrate how these operations arise.

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 an Identity Matrix [08/09/1999]
Which of the following would be an identity matrix?

Definition of an N-tuple [09/20/1999]
What is an n-tuple?

Describing Eigenvalues and Eigenvectors [04/05/1998]
What is an eigenvalue? an eigenvector? What are some applications?

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

Determinant of an Invertible Matrix [08/04/1999]
Could you explain and give a proof for why a matrix is invertible if and only if the determinant is non-zero?

Determinants and the Area of a Triangle [12/14/1998]
Given a triangle with vertices (A,B), (C,D), and (E,F), how do you find the area in determinant form?

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

Determining If A Polygon's Edges Cross [10/29/2003]
Is there a way to determine if a polygon has edges that cross by using the coordinates of the vertices that define the polygon [(x1,y1) to (xn,yn)]?

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)).

Distance Between 2 Lines: Vectors [8/19/1996]
What is the shortest distance between 2 lines?

Distance Between a Line and a Point Using Vectors [09/25/1999]
How do you use scalar vector projections to prove that the distance between the line ax+by+c = 0 and point (x1,y1) is |ax1+by1+c|/sqrt(a^2+b^2)?

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

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?

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