Linear Algebra

• Definition of a Vector [Jim, 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 [Toub, 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 Linear Algebra? [Choy, 09/06/1997]
  I have absolutely no idea about linear algebra...

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

• Rotating a cube in space [Bentz, 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?

• Linear Programming Problem-Maximizing Tents Produced [Ching, 10/6/1995]
  A manufacturer of a lightweight mountain tent makes a standard model and an expedition model. I know the labor hours needed for each by the cutting and assembly departments, the maximum labor hours available per week in each department, the distributor, will not take more than 12 expedition tents per week. If the company makes a profit of $50 on each standard tent and $80 on each expedition tent, how many tents of each type should be manufactured each week to maximize the weekly profit?

• Rearranging Matrices [Schmidt, 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.

• Laws of Vectors [Rovers, 4/10/1996]
  Question - Part 1: Use definitions I and II below to prove that: k[(a,b) + (c,d)] = k(a,b) + k(c,d)

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

• Powers of Matrices [Miguel, 6/27/1996]
  Would you send me some information/links about matrices and exponentials?

• Two Equations, Two Unknowns [Pooh, 7/27/1996]
  How do I solve for x and y in terms of a and b for the equations x^4 + y^4 = b^4 and x + y = a ?

• Solve for x, y, and z [Tirojava, 7/27/1996]
  How do I solve for x, y, and z in these three equations: 1/2y + 1/3z = 26, 1/3x + 1/4z = 23, and 1/2x + 1/4y = 28?

• Matrix Questions: Addition, Inverse [emsmdt06, 8/6/1996]
  (1 2) + (-1 -2) =? and if A = (3 4), does A inverse exist?

• Sum of Two Vectors [Kruger, 9/10/1996]
  Why is the magnitude of the sum of two vectors less than or equal to the sum of the magnitudes of each vector?

• Values of Scalers [Rudmanla, 9/11/1996]
  Can a scalar quantity be negative?

• Finding Solutions to Systems of Equations [Sheehy, 10/18/1996]
  I need help solving systems of equations, i.e. 3m+5n=21 and 4m+10n=38.

• Linear Systems [Farley, 10/22/1996]
  I need to solve this system of equations for x and y in terms of a, b, c, d, e, and f: ax + by = e; cx + dy = f.

• Matrices and Computer Programming [Robert, 11/05/1996]
  Why are matrix multiplication and matrix algebra the way they are?

• Solving Systems Using Augmented Matrices [Emmons, 11/05/1996]
  How do you solve a set of linear equations using an augmented matrix?

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

• Raytrace of a Star Sapphire [Charles, 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?

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

• Shrunk Axes [Jen, 01/20/1997]
  How do you use the distance formula when the coordinate axes are not perpendicular?

• Redundant Sets of Equations [Crowder, 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?

• Matrices and TI Calculators [Nichols, 03/06/1997]
  How do I solve a set of three equations in three unknowns using matrices on my TI-82 calculator?

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

• 3-D Points to a 2-D computer screen [Savage, 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...

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

• What is Nonlinear Math? [Jones, 10/02/1997]
  What exactly is nonlinear math, and what is it used for?

• What is a Loop? [Gardner, 10/06/1997]
  Further, exactly what is _an_ algebra? Thirdly, what is a Lie group?

• Types of Linear Systems [Devenell, 10/13/1997]
  What is meant when a type of linear system is said to be a consistent or an inconsistent system?

• Linear Algebra - Definition of Orthonormal [Wright, 10/28/1997]
  Show that the transition matrix P from one orthonormal basis to another is unitary, that is, P*P = I.

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

• Explaining the Determinant [Carlino, 11/16/1997]
  I am trying to understand what the determinant of a matrix actually is.

• 3D Geometry [Fallon, 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.

• Inverse of a Matrix [Sivaram, 12/05/1997]
  I need an algorithm to compute the inverse of a matrix.

• Vector Spaces [Dumitrescu, 12/09/1997]
  Why does C^2 considered as a vector space over the complex numbers have dimension 2, but as a vector space over the real numbers have dimension 4?

• Principal Axis Theorem [Goga, 12/10/1997]
  Could you give me applications of the principal axis theorem?

• Linear Equations; Rates of Pay [Michael, 12/16/1997]
  Five equation sets; and: A carpenter and his assistant working together for 8 hours earn a total of $96..

• Inverse of a Matrix [Catanzaro, 12/29/1997]
  How do you figure out the inverse of a 2 x 2 matrix?

• Solving 3x3 Matrices Manually [Miranda, 01/27/1998]
  How can I find the inverse of a 3x3 matrix manually (without pushing the inverse button on my calculator)?

• Shortest Distance between Points [Justin, 01/17/1998]
  I am doing a project on the shortest distance between two points via another plane. I need help with my theorems.

• Hilbert Space [Anderson, 02/04/1998]
  I'm doing reseach on quantum computations. Could you please tell me what Hilbert Space is?

• Solving a 6x6 System of Equations [Goodspeed, 03/24/1998]
  Is there a Cramer's rule for a 6x6 equation?

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

• Matrix Inverses [Hornbuckle, 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?

• Row-Reducing a Matrix [Audrey, 04/10/1998]
  Could you please teach me how to row-reduce a matrix?

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

• Using Determinants to Find Area [Goldner, 06/09/1998]
  How can you find the area of a triangle or a parallelogram using determinants?

• Linear Equations and Matrices [Lawson, 07/22/1998]
  I have a system of five linear equations, and I need to solve them using matrices.

• Vectors of Parallelograms and Octagons [Li, 07/28/1998]
  ABCDEFGH is a regular octagon and AB = p and BC = q. Express AH in terms of p and q...

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

• Yacht Distances and Vectors [Stokes, 09/05/1998]
  Can you help me find the smallest distance between two yachts using vectors?

• Inverse and Adjugate Matrices [Easton, 09/21/1998]
  How can you find the inverse of a matrix using the adjugate matrix? How would you then solve a system of three equations and three unknowns?

• The Intersection of Two Subspaces [Farber, 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?

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

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

• System of Equations and Gauss-Jordan [Jones, 11/29/1998]
  Solve using the Gauss-Jordan method: a 5-percent solution of a drug is mixed with 15- and 10-percent solutions...

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

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

• Gauss' Method for Solving Equations [Jacob, 12/11/1998]
  How do you use Gauss' method and matrices to solve systems of equations? Why does this method work?

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

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

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

• Solving Systems of Equations Using Matrices [Brian, 04/02/1999]
  Solve the following system of equations: a*(1-a) = b*c, b*(a+d-1) = 0, c*(a+d-1) = 0, d*(1-d) = b*c.

• Proof About Matrices [Hagiwara, 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?

• Vector Spaces [Reid, 06/11/1999]
  How can I tell if V = {(x,y) in R^2 | y = 3x+1} a vector space when addition and scalar multiplication are defined by (x,y) + (x',y') = (x+x',y+y'-1) and k(x,y) = (kx,k(y-1)+1)?

• Row Operations and an Augmented Matrix [Tomberlin, 07/26/1999]
  How can I solve a system of five simultaneous equations using row operations on the augmented matrix of the system?

• Three-Dimensional Cross Product Derivation [Wood, 07/26/1999]
  How can we prove the equation for 3-dimensional cross products in terms of the determinants a x (b x c) = (a.c)b - (a.b)c is correct?

• Finding Area Using Determinants [Victoria, 07/27/1999]
  Why are 1's put in the last column of the matrix when using the determinant to find the area of a region?

• Determinant of an Invertible Matrix [Smith, 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?

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

• Finding the Intersection of Two Circles [Groff, 08/19/1999]
  How can I find the intersection points of two circles?

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

• Distance Between a Line and a Point Using Vectors [Bui, 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)?

• Expansion by Minors [Bullard, 10/26/1999]
  How do you evaluate the determinant of a 3x3 matrix using expansion by minors?

• Symmetric Matrices [Marnie, 10/27/1999]
  Find all values of a, b, and c for which a 3x3 matrix A is symmetric, where the elements of A contain algebraic expressions in a, b, and c.

• Rotations in Three Dimensions [Drew, 11/11/1999]
  How can I rotate something about an axis or about a line in 3D?

• Using an Augmented Matrix [Katie, 11/21/1999]
  Solve the set of simultaneous equations x+3y+z=3, x+5y+5z=1 and 2x+6y+ 3z=8 using an augmented matrix?

• Equation of a Line in Three or More Dimensions [Jarosch, 05/18/2000]
  Can the equation y = mx + b be used to define a line in three dimensions? What about four or more dimensions?

• Equation for Angle Formed by Two Vectors [Michelle, 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?

• Properties of Determinants [Nooni, 10/23/2000]
  What properties of determinants do we use to evaluate them, and how? How do we put determinants in row-echelon form?

• The Stationary Vector [Searing, 11/13/2000]
  In Markov processes, what does a stationary vector tell you and how do you find it?

• An Absorbing States Problem [Goulding, 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?

• Big-O Notation in Matrix Multiplication [D'Amore, 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?

• Linearly Independent Set Proof [Golban, 01/24/2001]
  Assume that in a vector space V, the vectors u and v are linearly independent. Prove that the set {2u-v, u+5v} is linearly independent.

• Subspace [Anthony, 01/25/2001]
  Determine whether the following is a subspace of R^3...

• Understanding Vector Spaces [Matea, 03/14/2001]
  What does a vector space in R^n mean? How can I prove that a list of numbers is a vector space?

• Maximum Difference, Longitude and Latitude [Mini, 04/10/2001]
  Find the maximum longitude and latitude difference between two points on Earth 1000 kilometers apart.

• Fourth-Degree Polynomial [Miller, 04/26/2001]
  How can I construct a fourth-degree polynomial that contains all of the following points: (-1,2), (1,-2), (3,7), (2,1) and (0,3)?

• Ball Bouncing off a Line Segment [Robert, 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?

• Irreducible Polynomials [McLoone, 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').

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

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

• Prove Choice of Scalars Unique [Mary, 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.

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

• Two Dogs Pulling a Sled [Mike, 10/15/2001]
  What are the magnitude and direction of the force the person will have to exert on the sled to keep it moving along the centerline of the road at constant velocity?

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

• Invertible Matrices [Julie, 12/07/2001]
  Show that if A and B are nxn invertible matrices, then A^(-1)=(A+B)^(- 1)+(A+AB^(-1)A)^(-1).

• Volume of a Tetrahedron [Andrew, 01/23/2002]
  The volume of a tetrahedron is one-third the distance from a vertex to the opposite face, times the area of that face. Find a formula for the volume of a tetrahedron in terms of the coordinates of its vertices P, Q, R, and S.

• Adding Angles [Maes, 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.