Course: Linear Algebra Dnatube
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Lec 1 - Introduction to matrices
Introduction to matrices What a matrix is. How to add and subtract them.
Lec 2 - Matrix multiplication (part 1)
Matrix multiplication (part 1) Multiplying two 2x2 matrices.
Lec 3 - Matrix multiplication (part 2)
Matrix multiplication (part 2) More on multiplying matrices.
Lec 4 - Inverse Matrix (part 1)
Inverse Matrix (part 1) Taking the inverse of a 2x2 matrix
Lec 5 - Inverting matrices (part 2)
Inverting matrices (part 2) Inverting a 3x3 matrix
Lec 6 - Inverting Matrices (part 3)
Inverting Matrices (part 3) Using Gauss-Jordan elimination to invert a 3x3 matrix.
Lec 7 - Matrices to solve a system of eq ...
Matrices to solve a system of equations Using the inverse of a matrix to solve a system of equations.
Lec 8 - Matrices to solve a vector combi ...
Matrices to solve a vector combination problem Using matrices to figure out if some combination of 2 vectors can create a 3rd vector
Lec 9 - Singular Matrices
Singular Matrices When and why you can't invert a matrix.
Lec 10 - 3- variable linear equations ( ...
3-variable linear equations (part 1) Visual intuition of a 3-variable linear equation.
Lec 11 - Solving 3 Equations with 3 Unknowns
Solving 3 Equations with 3 Unknowns Solving 3 equations with 3 unknowns (old video from July 2008)
Lec 12 - Linear Algebra: Introduction to ...
Linear Algebra: Introduction to Vectors Linear Algebra: Introduction to Vectors
Lec 13 - Linear Algebra: Vector Examples
Linear Algebra: Vector Examples Visually understanding basic vector operations
Lec 14 - Linear Algebra: Parametric Repr ...
Linear Algebra: Parametric Representations of Lines Linear Algebra: Parametric Representations of Lines in R2 and R3
Lec 15 - Linear Combinations and Span
Linear Combinations and Span Understanding linear combinations and spans of vectors
Lec 16 - Linear Algebra: Introduction to ...
Linear Algebra: Introduction to Linear Independence Linear Algebra: Introduction to linear dependence and independence
Lec 17 - More on linear independence
More on linear independence More examples determining linear dependence or independence.
Lec 18 - Span and Linear Independence Ex ...
Span and Linear Independence Example Determining whether 3 vectors are linearly independent and/or span R3
Lec 19 - Linear Subspaces
Linear Subspaces Introduction to linear subspaces of Rn
Lec 20 - Linear Algebra: Basis of a Subspace
Linear Algebra: Basis of a Subspace Understanding the definition of a basis of a subspace
Lec 21 - Vector Dot Product and Vector L ...
Vector Dot Product and Vector Length Definitions of the vector dot product and vector length
Lec 22 - Proving Vector Dot Product Prop ...
Proving Vector Dot Product Properties Proving the "associative", "distributive" and "commutative" properties for vector dot products.
Lec 23 - Proof of the Cauchy-Schwarz Ine ...
Proof of the Cauchy-Schwarz Inequality Proof of the Cauchy-Schwarz Inequality
Lec 24 - Linear Algebra: Vector Triangle ...
Linear Algebra: Vector Triangle Inequality Proving the triangle inequality for vectors in Rn
Lec 25 - Defining the angle between vectors
Defining the angle between vectors Introducing the idea of an angle between two vectors
Lec 26 - Defining a plane in R3 with a p ...
Defining a plane in R3 with a point and normal vector Determining the equation for a plane in R3 using a point on the plane and a normal vector
Lec 27 - Linear Algebra: Cross Product I ...
Linear Algebra: Cross Product Introduction Introduction to the cross product
Lec 28 - Proof: Relationship between cro ...
Proof: Relationship between cross product and sin of angle Proof: Relationship between the cross product and sin of angle between vectors
Lec 29 - Dot and Cross Product Compariso ...
Dot and Cross Product Comparison/Intuition Dot and Cross Product Comparison/Intuition
Lec 30 - Matrices: Reduced Row Echelon F ...
Matrices: Reduced Row Echelon Form 1 Solving a system of linear equations by putting an augmented matrix into reduced row echelon form
Lec 31 - Matrices: Reduced Row Echelon F ...
Matrices: Reduced Row Echelon Form 2 Another example of solving a system of linear equations by putting an augmented matrix into reduced row echelon form
Lec 32 - Matrices: Reduced Row Echelon F ...
Matrices: Reduced Row Echelon Form 3 And another example of solving a system of linear equations by putting an augmented matrix into reduced row echelon form
Lec 33 - Matrix Vector Products
Matrix Vector Products Defining and understanding what it means to take the product of a matrix and a vector
Lec 34 - Introduction to the Null Space ...
Introduction to the Null Space of a Matrix Showing that the Null Space of a Matrix is a valid Subspace
Lec 35 - Null Space 2: Calculating the n ...
Null Space 2: Calculating the null space of a matrix Calculating the null space of a matrix
Lec 36 - Null Space 3: Relation to Linea ...
Null Space 3: Relation to Linear Independence Understanding how the null space of a matrix relates to the linear independence of its column vectors
Lec 37 - Column Space of a Matrix
Column Space of a Matrix Introduction to the column space of a matrix
Lec 38 - Null Space and Column Space Basis
Null Space and Column Space Basis Figuring out the null space and a basis of a column space for a matrix
Lec 39 - Visualizing a Column Space as a ...
Visualizing a Column Space as a Plane in R3 Determining the planar equation for a column space in R3
Lec 40 - Proof: Any subspace basis has s ...
Proof: Any subspace basis has same number of elements Proof: Any subspace basis has same number of elements
Lec 41 - Dimension of the Null Space or ...
Dimension of the Null Space or Nullity Dimension of the Null Space or Nullity
Lec 42 - Dimension of the Column Space o ...
Dimension of the Column Space or Rank Dimension of the Column Space or Rank
Lec 43 - Showing relation between basis ...
Showing relation between basis cols and pivot cols Showing that linear independence of pivot columns implies linear independence of the corresponding columns in the original equation
Lec 44 - Showing that the candidate basi ...
Showing that the candidate basis does span C(A) Showing that just the columns of A associated with the pivot columns of rref(A) do indeed span C(A).
Lec 45 - A more formal understanding of ...
A more formal understanding of functions A more formal understanding of functions
Lec 46 - Vector Transformations
Vector Transformations Introduction to the notion of vector transformations
Lec 47 - Linear Transformations
Linear Transformations Introduction to linear transformations
Lec 48 - Matrix Vector Products as Linea ...
Matrix Vector Products as Linear Transformations Matrix Vector Products as Linear Transformations
Lec 49 - Linear Transformations as Matri ...
Linear Transformations as Matrix Vector Products Showing how ANY linear transformation can be represented as a matrix vector product
Lec 50 - Image of a subset under a trans ...
Image of a subset under a transformation Exploring what happens to a subset of the domain under a transformation
Lec 51 - im(T): Image of a Transformation
im(T): Image of a Transformation Showing that the image of a subspace under a transformation is also a subspace. Definition of the image of a Transformation.
Lec 52 - Preimage of a set
Preimage of a set Definition of preimage of a set
Lec 53 - Preimage and Kernel Example
Preimage and Kernel Example Example involving the preimage of a set under a transformation. Definition of kernel of a transformation.
Lec 54 - Sums and Scalar Multiples of Li ...
Sums and Scalar Multiples of Linear Transformations Sums and Scalar Multiples of Linear Transformations. Definitions of matrix addition and scalar multiplication.
Lec 55 - More on Matrix Addition and Sca ...
More on Matrix Addition and Scalar Multiplication More on Matrix Addition and Scalar Multiplication
Lec 56 - Linear Transformation Examples: ...
Linear Transformation Examples: Scaling and Reflections Creating scaling and reflection transformation matrices (which are diagonal)
Lec 57 - Linear Transformation Examples: ...
Linear Transformation Examples: Rotations in R2 Linear Transformation Examples: Rotations in R2
Lec 58 - Rotation in R3 around the X-axis
Rotation in R3 around the X-axis Construction a rotation transformation in R3
Lec 59 - Unit Vectors
Unit Vectors What unit vectors are and how to construct them
Lec 60 - Introduction to Projections
Introduction to Projections Determining the projection of a vector on s line
Lec 61 - Expressing a Projection on to a ...
Expressing a Projection on to a line as a Matrix Vector prod Expressing a Projection on to a line as a Matrix Vector prod
Lec 62 - Compositions of Linear Transfor ...
Compositions of Linear Transformations 1 Introduction to compositions of Linear Transformations
Lec 63 - Compositions of Linear Transfor ...
Compositions of Linear Transformations 2 Providing the motivation for definition of matrix products
Lec 64 - Linear Algebra: Matrix Product ...
Linear Algebra: Matrix Product Examples Example of taking the product of two matrices
Lec 65 - Matrix Product Associativity
Matrix Product Associativity Showing that matrix products are associative
Lec 66 - Distributive Property of Matrix ...
Distributive Property of Matrix Products Showing that matrix products exhibit the distributive property
Lec 67 - Linear Algebra: Introduction to ...
Linear Algebra: Introduction to the inverse of a function Introduction to the inverse of a function
Lec 68 - Proof: Invertibility implies a ...
Proof: Invertibility implies a unique solution to f(x)=y Proof: Invertibility implies a unique solution to f(x)=y for all y in co-domain of f.
Lec 69 - Surjective (onto) and Injective ...
Surjective (onto) and Injective (one-to-one) functions Introduction to surjective and injective functions
Lec 70 - Relating invertibility to being ...
Relating invertibility to being onto and one-to-one Relating invertibility to being onto (surjective) and one-to-one (injective)
Lec 71 - Determining whether a transform ...
Determining whether a transformation is onto Determining whether a transformation is onto
Lec 72 - Linear Algebra: Exploring the s ...
Linear Algebra: Exploring the solution set of Ax=b Exploring the solution set of Ax=b (non homogeneous equations)
Lec 73 - Linear Algebra: Matrix conditio ...
Linear Algebra: Matrix condition for one-to-one trans Showing that the rank of the of an mxn transformation matrix has to be an for the transformation to be one-to-one (injective)
Lec 74 - Linear Algebra: Simplifying con ...
Linear Algebra: Simplifying conditions for invertibility Showing that a transformation is invertible if and only if rref(A) is equal to the identity matrix
Lec 75 - Linear Algebra: Showing that In ...
Linear Algebra: Showing that Inverses are Linear Showing that inverse transformations are also linear
Lec 76 - Linear Algebra: Deriving a meth ...
Linear Algebra: Deriving a method for determining inverses Determining a method for constructing inverse transformation matrices
Lec 77 - Linear Algebra: Example of Find ...
Linear Algebra: Example of Finding Matrix Inverse Example of calculating the inverse of a matrix
Lec 78 - Linear Algebra: Formula for 2x2 ...
Linear Algebra: Formula for 2x2 inverse Figuring out the formula for a 2x2 matrix. Defining the determinant.
Lec 79 - Linear Algebra: 3x3 Determinant
Linear Algebra: 3x3 Determinant Determinants: Finding the determinant of a 3x3 matrix
Lec 80 - Linear Algebra: nxn Determinant
Linear Algebra: nxn Determinant Defining the determinant for nxn matrices. An exampled of a 4x4 determinant.
Lec 81 - Linear Algebra: Determinants al ...
Linear Algebra: Determinants along other rows/cols Finding the determinant by going along other rows or columns
Lec 82 - Linear Algebra: Rule of Sarrus ...
Linear Algebra: Rule of Sarrus of Determinants A alternative "short cut" for calculating 3x3 determinants (Rule of Sarrus)
Lec 83 - Linear Algebra: Determinant whe ...
Linear Algebra: Determinant when row multiplied by scalar The determinant when a row is multiplied by a scalar
Lec 84 - Linear Algebra: (correction) sc ...
Linear Algebra: (correction) scalar muliplication of row Correction of last video showing that the determinant when one row is multiplied by a scalar is equal to the scalar times the determinant
Lec 85 - Linear Algebra: Determinant whe ...
Linear Algebra: Determinant when row is added The determinant when one matrix has a row that is the sum of the rows of other matrices (and every other term is identical in the 3 matrices)
Lec 86 - Linear Algebra: Duplicate Row D ...
Linear Algebra: Duplicate Row Determinant Determinant of a matrix with duplicate rows
Lec 87 - Linear Algebra: Determinant aft ...
Linear Algebra: Determinant after row operations What happens to the determinant when we perform a row operation
Lec 88 - Linear Algebra: Upper Triangula ...
Linear Algebra: Upper Triangular Determinant The determinant of an upper triangular matrix
Lec 89 - Linear Algebra: Simpler 4x4 det ...
Linear Algebra: Simpler 4x4 determinant Calculating a 4x4 determinant by putting in in upper triangular form first.
Lec 90 - Linear Algebra: Determinant an ...
Linear Algebra: Determinant and area of a parallelogram Realizing that the determinant of a 2x2 matrix is equal to the area of the parallelogram defined by the column vectors of the matrix
Lec 91 - Linear Algebra: Determinant as ...
Linear Algebra: Determinant as Scaling Factor Viewing the determinant of the transformation matrix as a scaling factor of regions
Lec 92 - Linear Algebra: Transpose of a ...
Linear Algebra: Transpose of a Matrix Transpose of a matrix
Lec 93 - Linear Algebra: Determinant of ...
Linear Algebra: Determinant of Transpose Proof by induction that transposing a matrix does not change its determinant
Lec 94 - Linear Algebra: Transpose of a ...
Linear Algebra: Transpose of a Matrix Product Taking the transpose of the product of two matrices
Lec 95 - Linear Algebra: Transposes of ...
Linear Algebra: Transposes of sums and inverses Transposes of sums and inverses
Lec 96 - Linear Algebra: Transpose of a ...
Linear Algebra: Transpose of a Vector Transpose of a column vector. Matrix-matrix products using vectors
Lec 97 - Linear Algebra: Rowspace and Le ...
Linear Algebra: Rowspace and Left Nullspace Rowspace and Left Nullspace
Lec 98 - Lin Alg: Visualizations of Left ...
Lin Alg: Visualizations of Left Nullspace and Rowspace Relationship between left nullspace, rowspace, column space and nullspace.
Lec 99 - Linear Algebra: Orthogonal Comp ...
Linear Algebra: Orthogonal Complements Orthogonal Complements as subspaces.
Lec 100 - Linear Algebra: Rank(A) = Rank ...
Linear Algebra: Rank(A) = Rank(transpose of A) Rank(A) = Rank(transpose of A)
Lec 101 - Linear Algebra: dim(V) + dim(o ...
Linear Algebra: dim(V) + dim(orthogonoal complelent of V)=n Showing that if V is a subspace of Rn, then dim(V) + dim(V's orthogonal complement) = n
Lec 102 - Lin Alg: Representing vectors ...
Lin Alg: Representing vectors in Rn using subspace members Showing that any member of Rn can be represented as a unique sum of a vector in subspace V and a vector in the orthogonal complement of V.
Lec 103 - Lin Alg: Orthogonal Complemen ...
Lin Alg: Orthogonal Complement of the Orthogonal Complement Finding that the orthogonal complement of the orthogonal complement of V is V
Lec 104 - Lin Alg: Orthogonal Complement ...
Lin Alg: Orthogonal Complement of the Nullspace The orthogonal complement of the nullspace and left nullspace.
Lec 105 - Lin Alg: Unique rowspace solut ...
Lin Alg: Unique rowspace solution to Ax=b Showing that, for any b that is in the column space of A, there is a unique member of the row space that is the "smallest" solution to Ax=b
Lec 106 - Linear Alg: Rowspace Solution ...
This is a lecture of Khan Academy. Visualizing the rowspace solution to Ax=b
Lec 107 - Linear Alg: Rowspace Solution ...
Linear Alg: Rowspace Solution to Ax=b example Visualizing the rowspace solution to Ax=b
Lec 108 - Lin Alg: Showing that A-transp ...
Lin Alg: Showing that A-transpose x A is invertible Showing that (transpose of A)(A) is invertible if A has linearly independent columns
Lec 109 - Linear Algebra: Projections on ...
Linear Algebra: Projections onto Subspaces Projections onto subspaces
Lec 110 - Linear Alg: Visualizing a proj ...
Linear Alg: Visualizing a projection onto a plane Visualizing a projection onto a plane. Showing that the old and new definitions of projections aren't that different.
Lec 111 - Lin Alg: A Projection onto a S ...
Lin Alg: A Projection onto a Subspace is a Linear Transforma Showing that a projection onto a subspace is a linear transformation
Lec 112 - Linear Algebra: Subspace Proje ...
Linear Algebra: Subspace Projection Matrix Example Example of a transformation matrix for a projection onto a subspace
Lec 113 - Lin Alg: Another Example of a ...
Lin Alg: Another Example of a Projection Matrix Figuring out the transformation matrix for a projection onto a subspace by figuring out the matrix for the projection onto the subspace's orthogonal complement first
Lec 114 - Linear Alg: Projection is clos ...
Linear Alg: Projection is closest vector in subspace Showing that the projection of x onto a subspace is the closest vector in the subspace to x
Lec 115 - Linear Algebra: Least Squares ...
Linear Algebra: Least Squares Approximation The least squares approximation for otherwise unsolvable equations
Lec 116 - Linear Algebra: Least Squares ...
Linear Algebra: Least Squares Examples An example using the least squares solution to an unsolvable system
Lec 117 - Linear Algebra: Another Least ...
Linear Algebra: Another Least Squares Example Using least squares approximation to fit a line to points
Lec 118 - Linear Algebra: Coordinates wi ...
Linear Algebra: Coordinates with Respect to a Basis Understanding alternate coordinate systems
Lec 119 - Linear Algebra: Change of Basi ...
Linear Algebra: Change of Basis Matrix Using a change of basis matrix to get us from one coordinate system to another.
Lec 120 - Lin Alg: Invertible Change of ...
Lin Alg: Invertible Change of Basis Matrix Using an invertible change of basis matrix to go between different coordinate systems
Lec 121 - Lin Alg: Transformation Matrix ...
Lin Alg: Transformation Matrix with Respect to a Basis Finding the transformation matrix with respect to a non-standard basis
Lec 122 - Lin Alg: Alternate Basis Tranf ...
Lin Alg: Alternate Basis Tranformation Matrix Example Example of finding the transformation matrix for an alternate basis
Lec 123 - Lin Alg: Alternate Basis Tranf ...
Lin Alg: Alternate Basis Tranformation Matrix Example Part 2 Showing that the transformation matrix with respect to basis B actually works. Brief point on why someone would want to operate in a different basis to begin with.
Lec 124 - Lin Alg: Changing coordinate s ...
Lin Alg: Changing coordinate systems to help find a transformation matrix Changing our coordinate system to find the transformation matrix with respect to standard coordinates
Lec 125 - Linear Algebra: Introduction t ...
Linear Algebra: Introduction to Orthonormal Bases Looking at sets and bases that are orthonormal -- or where all the vectors have length 1 and are orthogonal to each other.
Lec 126 - Linear Algebra: Coordinates wi ...
Linear Algebra: Coordinates with respect to orthonormal bases Seeing that orthonormal bases make for good coordinate systems
Lec 127 - Lin Alg: Projections onto subs ...
Lin Alg: Projections onto subspaces with orthonormal bases Projections onto subspaces with orthonormal bases
Lec 128 - Lin Alg: Finding projection on ...
Lin Alg: Finding projection onto subspace with orthonormal basis example Example of finding the transformation matrix for the projection onto a subspace with an orthonormal basis
Lec 129 - Lin Alg: Example using orthogo ...
Lin Alg: Example using orthogonal change-of-basis matrix to find transformation matrix Example using orthogonal change-of-basis matrix to find transformation matrix
Lec 130 - Lin Alg: Orthogonal matrices ...
Lin Alg: Orthogonal matrices preserve angles and lengths Showing that orthogonal matrices preserve angles and lengths
Lec 131 - Linear Algebra: The Gram-Schmi ...
Linear Algebra: The Gram-Schmidt Process Finding an orthonormal basis for a subspace using the Gram-Schmidt Process
Lec 132 - Linear Algebra: Gram-Schmidt P ...
Linear Algebra: Gram-Schmidt Process Example Using Gram-Schmidt to find an orthonormal basis for a plane in R3
Lec 133 - Linear Algebra: Gram-Schmidt ...
Linear Algebra: Gram-Schmidt example with 3 basis vectors Gram-Schmidt example with 3 basis vectors
Lec 134 - Linear Algebra: Introduction t ...
Linear Algebra: Introduction to Eigenvalues and Eigenvectors What eigenvectors and eigenvalues are and why they are interesting
Lec 135 - Linear Algebra: Proof of formu ...
Linear Algebra: Proof of formula for determining Eigenvalues Proof of formula for determining Eigenvalues
Lec 136 - Linear Algebra: Example solvin ...
Linear Algebra: Example solving for the eigenvalues of a 2x2 matrix Example solving for the eigenvalues of a 2x2 matrix
Lec 137 - Linear Algebra: Finding Eigen ...
Linear Algebra: Finding Eigenvectors and Eigenspaces example Finding the eigenvectors and eigenspaces of a 2x2 matrix
Lec 138 - Linear Algebra: Eigenvalues of ...
Linear Algebra: Eigenvalues of a 3x3 matrix Determining the eigenvalues of a 3x3 matrix
Lec 139 - Linear Algebra: Eigenvectors ...
Linear Algebra: Eigenvectors and Eigenspaces for a 3x3 matrix Eigenvectors and eigenspaces for a 3x3 matrix
Lec 140 - Linear Algebra: Showing that ...
Linear Algebra: Showing that an eigenbasis makes for good coordinate systems Showing that an eigenbasis makes for good coordinate systems
Lec 141 - Vector Triple Product Expansio ...
This lecture is provided by Khan Academy. A shortcut for having to evaluate the cross product of three vectors
Lec 142 - Normal vector from plane equation
This is a lecture of Khan Academy. Figuring out a normal vector to a plane from its equation
Lec 143 - Point distance to plane
This lecture is provided by Khan Academy. Distance between a point and a plane in three dimensions
Lec 144 - Distance Between Planes
This is a lecture of Khan Academy. Distance Between Planes 2010 IIT JEE Paper 1 Problem 51 Distance Between Planes
Linear Algebra
Source of these courses is khan
Matrices, vectors, vector spaces, transformations. Covers all topics in a first year college linear algebra course. This is an advanced course normally taken by science or engineering majors after taking at least two semesters of calculus (although calculus really isn't a prereq) so don't confuse this with regular high school algebra.
khan Website: http://www.dnatube.com/school/khan
