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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.

Finding an Angle between a Line and a Line Segment [10/29/1995]
What is the best way to find the angle between a line and a line segment that both originate at the same point?

Finding an Orthonormal Matrix to Rotate an n-Dimensional Vector [08/05/2009]
Doctor George invokes Rodrigues' Rotation Formula and the Householder reflection to show how to build an orthonormal matrix M so that M * v1 = v2 for two n-dimensional vectors v1 and v2.

Finding Area Using Determinants [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?

Finding Eigenvalues and Eigenvectors [06/27/2007]
In a 3x3 square matrix, how would you find the eigenvalues and eigenvectors?

Finding Line of Best Fit with Matrices [06/30/2004]
How can I find the line of best fit for a set of data using matrices?

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

Finding the Angle between Two Vectors [01/15/2006]
Two vectors A and B are of equal magnitude. If the magnitude of A+B is 120 times larger than the magnitude of A-B, what is the angle between them?

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

Finding the Other End of a Line Segment [07/25/2002]
Given a point, the slope of a line segment, and a length, how do I find the other end of the line segment?

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?

Find the Velocity - a Vector Problem [09/25/2002]
A river flows westward at 8 m/s. A person wants to go directly across the river so that the resultant velocity is 12 m/s northward. Find the velocity of the motorboat that would be required to achieve this resultant velocity.

Fourth-Degree Polynomial [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)?

Gaussian Elimination (a 4x4) [11/29/2001]
I have a 4x4 problem: A+3B+C+3D=14; 4A-2B-3C+D=20; 2A+B-C-D=9; A+2B-C-2D=3.

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

The Haybaler Problem: Reflections and Refinements [10/05/2002]
Given pairwise weights, a student and Doctor Greenie reflect on how to deduce individual ones before a PhD follows up with several simplifying insights of his own.

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

Interesting Method of Finding a Determinant [08/01/2008]
I need to show that the right side is the determinant of the given matrix.

Intersecting Vectors and the Dot Product [04/24/1998]
Each of the following geometrical theorems can be proved with vectors, using the dot product...

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 and Adjugate Matrices [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?

Inverse Matrix [05/05/2003]
Use the inverse matrix to find values x, y, and z.

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

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

Inverse of a Square Matrix, Proven [08/01/2014]
A teen seeks proof of the formula for the inverse of a square matrix. Expanding by cofactors, Doctor Fenton provides an outline.

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').

Laws of Vectors [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)

Leading Variables, Free Variables, and the Coefficient Zero [01/23/2010]
What do you call a variable with a 0 in front of it? Doctor Vogler explains the terms leading variable and free variable with an example.

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.

Linear and Circular Parametric Equations [05/05/1998]
Using parametric equations to plot a line segment connecting any two points on the plane.

Linear Combinations of Vectors [10/23/2005]
Can every vector in the xy plane be written as a linear combination of the vectors u = (1,4) and v = (-2,5)? When can and can't one vector be written as a linear combination of two others?

Linear Dependence of Vectors [12/04/2008]
If the vectors (2,-1,4), (2,k+6,k), and (1,0,2) are linearly independent, then what is the value of k? Can k have more than one value?

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

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

Linearly Independent Set Proof [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.

Linear Programming Problem-Maximizing Tents Produced [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?

Linear Systems [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.

Lost in the Array of Operations When Multiplying Matrices [03/30/2010]
A student writes about her difficulty carrying out matrix multiplication. Doctor Ian suggests a way to systematically organize and step through the individual dot products and sums involved.

Magic Square Matrix [03/24/2003]
Let M be an integer-valued 3x3 matrix whose entries form a magic square. Let s be the sum of all entries in M and d be the determinant of M. Show that d/s is an integer.

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