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.
Course
Linear Algebra
56 Lectures
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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!
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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.
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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.
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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.
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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.
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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.
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Cramer's Rule to Solve a System of 3 Linear Equations - Example 1
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Cramer's Rule to Solve a System of 3 Linear Equations - Example 2
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Using Gauss-Jordan to Solve a System of Three Linear Equations - Example 1
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Using Gauss-Jordan to Solve a System of Three Linear Equations - Example 2
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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.
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Solving a Dependent System of Linear Equations involving 3 Variables
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Systems of Linear Equations - Inconsistent Systems Using Elimination by Addition - Example 1
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Systems of Linear Equations - Inconsistent Systems Using Elimination by Addition - Example 2
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Systems of Linear Equations - Inconsistent Systems Using Elimination by Addition - Example 3
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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!!
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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!!
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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!!
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Finding the Determinant of a 3 x 3 matrix. I show the basic formula and compute the determinant of a specific matrix.
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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!!!
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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!
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Solving Systems of Linear Equations Using Elimination By Addition - Two complete examples and part of a third problem are shown!
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Multiplying Matrices - Two examples of multiplying a matrix by another matrix are shown.
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Matrix Operations - Adding, Subtracting, and Multiplying by a constant for matrices is discussed.
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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.
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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.
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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.
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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!)
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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.
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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!).
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Finding a Unit Vector, Ex 2. In this video we find a unit vector associated with a given vector.
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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.
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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.
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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).
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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!).
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Magnitude and Direction of a Vector, Example 1. Here we find the magnitude (length) of some vectors and find the angle associated with them.
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Magnitude and Direction of a Vector, Example 2. Here we find the magnitude (length) of some vectors and find the angle associated with them.
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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.
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When are two vectors considered to be the same?
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An introduction to vectors: magnitude, direction, length, component form are all discussed.
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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.
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Vector Basics - Components, adding vectors algebraically and multiplying by a constant.
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Vector Basics - Components, adding vectors algebraically and multiplying by a constant. PART 2 of the video that got cut off!!
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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.
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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.
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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!
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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.
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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.
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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.
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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.
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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!
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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.
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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!
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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.
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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.