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!
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 - 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 - 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. 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. 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. 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. 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
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
Using Gauss-Jordan to Solve a System of Three Linear Equations - Example 2
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
Systems of Linear Equations - Inconsistent Systems Using Elimination by Addition - Example 1
Systems of Linear Equations - Inconsistent Systems Using Elimination by Addition - Example 2
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. 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. 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. 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. I show the basic formula and compute the determinant of a specific matrix.
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 - This is a follow up to Part 1 where the last example was cut off!
Solving Systems of Linear Equations Using Elimination By Addition - Two complete examples and part of a third problem are shown!
Multiplying Matrices - Two examples of multiplying a matrix by another matrix are shown.
Matrix Operations - Adding, Subtracting, and Multiplying by a constant for matrices is discussed.
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. 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. 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. 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. 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. 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. In this video we find a unit vector associated with a given vector.
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. 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. 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. 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. Here we find the magnitude (length) of some vectors and find the angle associated with them.
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. 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?
An introduction to vectors: magnitude, direction, length, component form are all discussed.
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 - Components, adding vectors algebraically and multiplying by a constant.
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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.