# Linear Algebra

PatrickJMT
• ##### The Span of a Set of Vectors
Patrick Jones

The Span of a Set of Vectors. In this video, I look at the notion of a span of a vector set. I work in R2 just to keep things simple, but the results can be generalized! I show how to justify that two vectors do in fact span all of R2.

• ##### Determinants to Find the Area of a Polygon
Patrick Jones

Determinants to Find the Area of a Polygon - In this video, I show how one can use determinants to find the area enclosed by any polygon!

• ##### Determinants to Find the Area of a Triangle
Patrick Jones

Determinants to Find the Area of a Triangle - In this video, I show how one can use determinants to find the area of a triangle.

• ##### Determinant of a 2 x 2 Matrix - A Few Basic Questions
Patrick Jones

Determinant of a 2 x 2 Matrix - A Few Basic Questions. In this video, I show how to find the determinant of a 2 x 2 matrix, and do a few related problems.

• ##### Finding the Inverse of a 3 x 3 Matrix using Determinants and Cofactors - Example 1
Patrick Jones

Finding the Inverse of a 3 x 3 Matrix using Determinants and Cofactors - Example 1. Besides using row reduction, this is another way to find the inverse of a 3 x 3 matrix.

• ##### Finding the Inverse of a 3 x 3 Matrix using Determinants and Cofactors - Example 2
Patrick Jones

Finding the Inverse of a 3 x 3 Matrix using Determinants and Cofactors - Example 2. Besides using row reduction, this is another way to find the inverse of a 3 x 3 matrix.

• ##### Finding the Inverse of a 3 x 3 Matrix using Determinants and Cofactors - Example 3
Patrick Jones

Finding the Inverse of a 3 x 3 Matrix using Determinants and Cofactors - Example 3. Besides using row reduction, this is another way to find the inverse of a 3 x 3 matrix.

• ##### Cramer's Rule to Solve a System of 3 Linear Equations - Example 1
Patrick Jones

Cramer's Rule to Solve a System of 3 Linear Equations - Example 1

• ##### Cramer's Rule to Solve a System of 3 Linear Equations - Example 2
Patrick Jones

Cramer's Rule to Solve a System of 3 Linear Equations - Example 2

• ##### Using Gauss-Jordan to Solve a System of Three Linear Equations - Example 1
Patrick Jones

Using Gauss-Jordan to Solve a System of Three Linear Equations - Example 1

• ##### Using Gauss-Jordan to Solve a System of Three Linear Equations - Example 2
Patrick Jones

Using Gauss-Jordan to Solve a System of Three Linear Equations - Example 2

• ##### Solving a 3 x 3 System of Equations Using the Inverse
Patrick Jones

Solving a 3 x 3 System of Equations Using the Inverse. In this video, I solve a system of three linear equations by using the inverse.

• ##### Solving a Dependent System of Linear Equations involving 3 Variables
Patrick Jones

Solving a Dependent System of Linear Equations involving 3 Variables

• ##### Systems of Linear Equations - Inconsistent Systems Using Elimination by Addition - Example 1
Patrick Jones

Systems of Linear Equations - Inconsistent Systems Using Elimination by Addition - Example 1

• ##### Systems of Linear Equations - Inconsistent Systems Using Elimination by Addition - Example 2
Patrick Jones

Systems of Linear Equations - Inconsistent Systems Using Elimination by Addition - Example 2

• ##### Systems of Linear Equations - Inconsistent Systems Using Elimination by Addition - Example 3
Patrick Jones

Systems of Linear Equations - Inconsistent Systems Using Elimination by Addition - Example 3

• ##### Solving a System of Equations Involving 3 Variables Using Elimination by Addition - Example 1
Patrick Jones

Solving a System of Equations Involving 3 Variables Using Elimination by Addition - Example 1. There are TONS of fractions, so I hope you do not get drowned out with all the arithmetic!!

• ##### Solving a System of Equations Involving 3 Variables Using Elimination by Addition - Example 2
Patrick Jones

Solving a System of Equations Involving 3 Variables Using Elimination by Addition - Example 2. There are TONS of fractions, so I hope you do not get drowned out with all the arithmetic!!

• ##### Solving a System of Equations Involving 3 Variables Using Elimination by Addition - Example 3
Patrick Jones

Solving a System of Equations Involving 3 Variables Using Elimination by Addition - Example 2. There are TONS of fractions, so I hope you do not get drowned out with all the arithmetic!!

• ##### Finding the Determinant of a 3 x 3 matrix
Patrick Jones

Finding the Determinant of a 3 x 3 matrix. I show the basic formula and compute the determinant of a specific matrix.

• ##### Row Reducing a Matrix - Systems of Linear Equations - Part 1
Patrick Jones

Row Reducing a Matrix - Systems of Linear Equations - Part 1. Basic notation and procedure as well as a full example are shown. The last part of the second part got cut off, but is finished in another video!!!

• ##### Row Reducing a Matrix - Systems of Linear Equations - Part 2
Patrick Jones

Row Reducing a Matrix - Systems of Linear Equations - Part 2 - This is a follow up to Part 1 where the last example was cut off!

• ##### Solving Systems of Equations Using Elimination By Addition
Patrick Jones

Solving Systems of Linear Equations Using Elimination By Addition - Two complete examples and part of a third problem are shown!

• ##### Multiplying Matrices - Example 1
Patrick Jones

Multiplying Matrices - Two examples of multiplying a matrix by another matrix are shown.

• ##### Matrix Operations
Patrick Jones

Matrix Operations - Adding, Subtracting, and Multiplying by a constant for matrices is discussed.

• ##### An Introduction to the Dot Product
Patrick Jones

An Introduction to the Dot Product. In this video, I give the formula for the dot product of two vectors, discuss the geometric meaning of the dot product, and find the dot product between some vectors.

• ##### Sketching Sums and Differences of Vectors
Patrick Jones

Sketching Sums and Differences of Vectors. In this video, I give three vectors, and do a few examples of sketching the sum and difference of those vectors.

• ##### Word Problems Involving Velocity or Other Forces (Vectors), Ex 1
Patrick Jones

Word Problems Involving Velocity or Other Forces (Vectors), Ex 1. In this problem we do a word problem involving the bearing (direction) of a boat.

• ##### Word Problems Involving Velocity or Other Forces (Vectors), Ex 2
Patrick Jones

Word Problems Involving Velocity or Other Forces (Vectors), Ex 2. In this problem we are given the bearing and velocity of a plane and the bearing and velocity of the wind; we want to find out the actual velocity of the plane after taking the wind into consideration. (a nice little problem!)

• ##### Word Problems Involving Velocity or Other Forces (Vectors), Ex 3.
Patrick Jones

Word Problems Involving Velocity or Other Forces (Vectors), Ex 3. Here we know the force required to keep a box from sliding down a ramp; we want to know the angle of inclination of the ramp.

• ##### Finding a Unit Vector, Ex 1
Patrick Jones

Finding a Unit Vector, Ex 1. In this video I discuss the idea of a unit vector and show how to find it (divide the vector by its magnitude!).

• ##### Finding a Unit Vector, Ex 2
Patrick Jones

Finding a Unit Vector, Ex 2. In this video we find a unit vector associated with a given vector.

• ##### Finding the Components of a Vector, Ex 1
Patrick Jones

Finding the Components of a Vector, Ex 1. In this video, we are given the magnitude and direction angle for the vector and want to express the vector in component form.

• ##### Finding the Components of a Vector, Ex 2
Patrick Jones

Finding the Components of a Vector, Ex 2. In this video, we are given the magnitude and direction angle for the vector and want to express the vector in component form.

• ##### Vector Addition and Scalar Multiplication, Example 1
Patrick Jones

Vector Addition and Scalar Multiplication, Example 1. In this video, we look at vector addition and scalar multiplication algebraically using the component form of the vector. I do not graph the vectors in this video (but do in others).

• ##### Vector Addition and Scalar Multiplication, Example 2
Patrick Jones

Vector Addition and Scalar Multiplication, Example 2. In this video I add two vectors in component form and also sketch the vectors to illustrate how to add vectors graphically (very useful stuff!).

• ##### Magnitude and Direction of a Vector, Example 1
Patrick Jones

Magnitude and Direction of a Vector, Example 1. Here we find the magnitude (length) of some vectors and find the angle associated with them.

• ##### Magnitude and Direction of a Vector, Example 2
Patrick Jones

Magnitude and Direction of a Vector, Example 2. Here we find the magnitude (length) of some vectors and find the angle associated with them.

• ##### Magnitude and Direction of a Vector, Example 3
Patrick Jones

Magnitude and Direction of a Vector, Example 3. Here we find the magnitude (length) of one last vector and find the angle associated with them.

• ##### When Are Two Vectors Considered to Be the Same?
Patrick Jones

When are two vectors considered to be the same?

• ##### An Introduction to Vectors, Part 1
Patrick Jones

An introduction to vectors: magnitude, direction, length, component form are all discussed.

• ##### Finding the Vector Equation of a Line
Patrick Jones

Finding the Vector Equation of a Line - In this video, I give the formula to find the vector equation of a line and do two examples.

• ##### Vector Basics - Algebraic Representations
Patrick Jones

Vector Basics - Components, adding vectors algebraically and multiplying by a constant.

• ##### Vector Basics - Algebraic Representations Part 2
Patrick Jones

Vector Basics - Components, adding vectors algebraically and multiplying by a constant. PART 2 of the video that got cut off!!

• ##### Vector Basics - Drawing Vectors/ Vector Addition
Patrick Jones

Vector Basics - Drawing Vectors/ Vector Addition. In this video, I discuss the basic notion of a vector, and how to add vectors together graphically as well as what it means graphically to multiply a vector by a scalar.

• ##### Vectors - The Dot Product
Patrick Jones

Vectors - The Dot Product. I show how to compute the dot product of two vectors, along with some useful theorems and results involving dot products. 3 complete examples are shown.

• ##### Vectors - Finding Magnitude or Length
Patrick Jones

Vectors - Finding Magnitude or Length. I give the formula, and do a couple examples of finding the magitude, or length, or a vector. Nothing heavy!

• ##### Linear Independence and Linear Dependence, Ex 1
Patrick Jones

Linear Independence and Linear Dependence, Ex 1. In this video, I explore the idea of what it means for a set of vectors to be linearly independent or dependent. I then work an example showing that a set of vectors is linearly dependent.

• ##### Linear Independence and Linear Dependence, Ex 2
Patrick Jones

Linear Independence and Linear Dependence, Ex 2. As a follow up to Ex 1, I show a set of vectors that is linearly independent by using row reduction.

• ##### Homogeneous Systems of Linear Equations - Trivial and Nontrivial Solutions, Part 1
Patrick Jones

Homogeneous Systems of Linear Equations - Trivial and Nontrivial Solutions, Part 1. In this video, I show what a homogeneous system of linear equations is, and show what it means to have only trivial solutions. In the next video, I work out an example that has nontrivial solutions.

• ##### Homogeneous Systems of Linear Equations - Trivial and Nontrivial Solutions, Part 2
Patrick Jones

Homogeneous Systems of Linear Equations - Trivial and Nontrivial Solutions, Part 2. In this video, I show how to find solutions to a homogeneous system of linear equations that has nontrivial solutions.

• ##### Useful Things to Remember About Linearly Independent Vectors
Patrick Jones

Useful Things to Remember About Linearly Independent Vectors. Just a few things that I think are useful to remember about linearly independent and dependent sets of vectors!

• ##### Basis for a Set of Vectors
Patrick Jones

Basis for a Set of Vectors. In this video, I give the definition for a apos; basis apos; of a set of vectors. I think proceed to work an example that shows three vectors that I picked form a basis for R_3.

• ##### Procedure to Find a Basis for a Set of Vectors
Patrick Jones

Procedure to Find a Basis for a Set of Vectors. In this video, I start with a set of vectors in R_3 and find a basis for those vectors. The basis is NOT necessarily unique!

• ##### Linear Transformations , Example 1, Part 1 of 2
Patrick Jones

Linear Transformations , Example 1, Part 1 of 2. In this video, I introduce the idea of a linear transformation of vectors from one space to another. I then proceed to show an example of whether or not a particular transformation is linear or not.

• ##### Linear Transformations , Example 1, Part 2 of 2
Patrick Jones

Linear Transformations , Example 1, Part 2 of 2. In this video, I introduce the idea of a linear transformation of vectors from one space to another. I then proceed to finish an example of whether or not a particular transformation is linear or not.