Geometric Vectors

Keywords: vector addition, scalar multiplication , zero vector, negative of a vector

• Introduction
• Representing geometric vectors
• Adding geometric vectors
• Multiplying geometric vectors by scalars
• Calculation rules for addition and scalar multiplication

---

Vectors in Coordinate Systems

Keywords: coordinates, components, i, j and k, position vector, norm, length, normalized vector

• Introduction
• Two-dimensional coordinate systems
• Three-dimensional coordinate systems
• Using components to calculate with vectors
• Calculating the length of a vector

---

Dot Products of Vectors

Keywords: dot product, angle between vectors, orthogonal vectors, orthogonal projections

• Introduction
• Two definitions of dot products
• Calculation rules for dot products
• Finding angles with dot products
• Orthogonal projections

---

Using Vectors to Describe Lines

Keywords: vector equation of a line, direction vector of a line, parametric equation of a line, symmetric equations of a line, uniform linear motion

• Introduction
• The vector form of a line
• The parametric form of a line
• The symmetric form of a line
• Finding lines
• Uniform linear motion

---

Using Vectors to Describe Planes

Keywords: point-normal equation of a plane, normal vector of a plane, standard equation of a plane, vector equation of a plane, parametric equations of a plane

• Introduction
• The point-normal form of a plane
• The standard equation of a plane
• The vector equation of a plane
• Finding planes

---

Vector Spans

Keywords: linear combination, span (noun), span (verb)

• Introduction
• Definition of a span
• The span of two vectors
• The span of three or more vectors

---

Linear Independence

Keywords: linearly independent vectors, linearly dependent vectors, linear dependence relation among vectors, trivial and non-trivial solutions

• Introduction
• The basic definition of linear independence
• Examples of linearly independent geometric vectors
• A more practical form of linear independence

---

Bases and Coordinate Systems

Keywords: basis, component, orthogonal basis, orthonormal basis

• Introduction
• Definition of a basis
• Basis-defined coordinate systems
• Orthogonal and orthonormal bases

---

Linear Operators on Geometric Vectors

Keywords: linear operator, rotation, reflection, projection, scale change, shear, standard matrix, matrix operator, orthogonal operator, orthogonal matrix

• Introduction
• Definition of a linear operator
• Examples of linear operators
• Linear operators in coordinates
• Orthogonal linear operators

---

A Catalogue of Simple Matrix Operators

Keywords: rotation, shear , reflection, scale, projection

• Introduction
• Projections in 2-space
• Reflections in 2-space
• Scale changes in 2-space
• Shears in 2-space
• Rotations in 2-space
• A sampler of 3-space matrix operators

---

Linear Systems and How to Solve Them

Keywords: linear system, solution to a linear system, augmented matrix, elementary row operations, reduced row-echelon form, row-echelon form, leading variables, back substitution

• Introduction
• Augmented matrices and row operations
• Solving a system in reduced row echelon form
• Solving a system in row echelon form

---

Matrices and Matrix Calculations

Keywords: matrix addition, zero matrix, negative of a matrix, matrix scalar multiplication, identity matrix, inverse of a matrix

• Introduction
• Addition, subtraction and scalar multiplication
• Multiplying matrices
• The inverse of a matrix and how to find it.

---

How to Row Reduce a Matrix

Keywords row operations, row reduction, reduced row echelon form, row echelon form, Gauss-Jordan elimination, Gaussian elimination

• Introduction
• Row operations and reduced row-echelon form
• Gauss-Jordan elimination
• Row reduction with a computer

---

The Formal Definition of a Determinant

Keywords: elementary product, determinant definition, diagonals method, triangular matrices, row operations, column operations

• Introduction
• The formal definition of a determinant
• Determinants of special matrices
• Using row and column operations on determinants

---

Minors, Cofactors and the Laplace Expansion of Determinants

Keywords: determinant, Laplace expansion, minor, cofactor, adjoint of a matrix, inverse of a matrix

• Introduction
• Minors and cofactors
• The Laplace expansion of a determinant
• Adjoints and inverses of matrices

1. Geometric Vectors
2. Vectors in Coordinate Systems
3. Dot Products of Vectors
4. Using Vectors to Describe Lines
5. Using Vectors to Describe Planes
6. Vector Spans
7. Linear Independence
8. Bases and Coordinate Systems
9. Linear Operators on Geometric Vectors
10. A Catalogue of Simple Matrix Operators
11. Linear Systems and How to Solve Them
12. Matrices and Matrix Calculations
13. How to Row Reduce a Matrix
14. The Formal Definition of a Determinant
15. Minors, Cofactors and the Laplace Expansion of Determinants

Voice version of Number 15: How to Row Reduce a Matrix