Vectors and spaces
17 videos
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Linear Algebra - Vectors and Spaces: What You Need to Know
Introduction to vectors and linear algebra fundamentals. Covers vectors, linear combinations, span, independence, and subspaces as the foundation for modern technology.
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Linear Algebra - What Is a Vector? Magnitude, Direction, and Rn
Learn what vectors are and how they differ from scalars. Vectors have both magnitude and direction, represented as arrows in coordinate spaces like R² and R³.
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Linear Algebra - Adding Vectors: Algebraically and Graphically
Learn vector addition through algebra and the tip-to-tail method. Add matching components to find the resultant vector, visualized by connecting vectors graphically.
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Linear Algebra - Scalar Multiplication: Stretching and Flipping Vectors
Learn scalar multiplication: how multiplying vectors by numbers stretches, shrinks, or flips them while preserving or reversing direction.
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Linear Algebra - Unit Vectors: Direction Without the Clutter
Learn what unit vectors are, how they represent pure direction with magnitude 1, and how to convert any vector into a unit vector by dividing by its magnitude.
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Linear Algebra - Finding a Unit Vector: Step-by-Step
Learn how to find a unit vector by dividing a vector by its magnitude. Step-by-step example shows how to normalize vector (3,4) to get direction without length.
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Linear Algebra - The Dot Product: Multiplying Vectors Into a Number
Learn how the dot product multiplies two vectors to produce a scalar that measures how aligned they are. Includes calculation examples and geometric interpretation.
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Linear Algebra - The Cross Product: A Vector Perpendicular to Both
Learn how the cross product creates a perpendicular vector from two 3D vectors, unlike the dot product which returns a scalar. Includes formula explanation and example.
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Linear Algebra - Linear Combinations and Span
Learn how linear combinations and span work in linear algebra. Discover how two non-collinear vectors can reach any point on a 2D plane using scalars.
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Linear Algebra - Linear Independence: Are Your Vectors Redundant?
This video explains linear dependence and independence of vectors using team-building analogy, showing how independent vectors add unique directions while dependent ones are redundant.
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Linear Algebra - Testing Linear Independence in R3
Learn how to test if three 3D vectors are linearly independent or dependent by solving a system of equations and checking for non-zero solutions.
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Linear Algebra - Subspaces: Spaces Within Spaces
Learn what subspaces are in linear algebra: subsets of vector spaces that satisfy closure under addition, scalar multiplication, and contain the zero vector.
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Linear Algebra - Basis and Dimension of a Subspace
Learn what a basis is in linear algebra: the minimal set of independent vectors that span a subspace. Discover how dimension relates to basis and why it's always constant.
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Linear Algebra - Null Space of a Matrix
Learn about the null space of a matrix: the set of all vectors that get mapped to zero. Includes a concrete example showing how to find and visualize the null space.
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Linear Algebra - Column Space and Rank of a Matrix
Learn what the column space of a matrix is, how to find it using row reduction, and how rank reveals the dimension of all possible outputs a matrix can produce.
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Linear Algebra - Solving Systems with Augmented Matrices and RREF
Learn to solve systems of linear equations using augmented matrices and row reduction to reach reduced row echelon form (RREF) for quick solutions.
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Linear Algebra - Equation of a Plane Using a Normal Vector
Learn how to describe a plane in 3D using a normal vector and a point. Understand how the plane equation ax+by+cz=d relates to perpendicular vectors.