Lec 38 - Null Space and Column Space Basis

Lecture list for this course

Lec 1 - Introduction to matrices

Lec 2 - Matrix multiplication (part 1)

Lec 3 - Matrix multiplication (part 2)

Lec 4 - Inverse Matrix (part 1)

Lec 5 - Inverting matrices (part 2)

Lec 6 - Inverting Matrices (part 3)

Lec 7 - Matrices to solve a system of equations

Lec 8 - Matrices to solve a vector combination problem

Lec 9 - Singular Matrices

Lec 10 - 3- variable linear equations (part 1)

Lec 11 - Solving 3 Equations with 3 Unknowns

Lec 12 - Linear Algebra: Introduction to Vectors

Lec 13 - Linear Algebra: Vector Examples

Lec 14 - Linear Algebra: Parametric Representations of Lines

Lec 15 - Linear Combinations and Span

Lec 16 - Linear Algebra: Introduction to Linear Independence

Lec 17 - More on linear independence

Lec 18 - Span and Linear Independence Example

Lec 19 - Linear Subspaces

Lec 20 - Linear Algebra: Basis of a Subspace

Lec 21 - Vector Dot Product and Vector Length

Lec 22 - Proving Vector Dot Product Properties

Lec 23 - Proof of the Cauchy-Schwarz Inequality

Lec 24 - Linear Algebra: Vector Triangle Inequality

Lec 25 - Defining the angle between vectors

Lec 26 - Defining a plane in R3 with a point and normal vector

Lec 27 - Linear Algebra: Cross Product Introduction

Lec 28 - Proof: Relationship between cross product and sin of angle

Lec 29 - Dot and Cross Product Comparison/Intuition

Lec 30 - Matrices: Reduced Row Echelon Form 1

Lec 31 - Matrices: Reduced Row Echelon Form 2

Lec 32 - Matrices: Reduced Row Echelon Form 3

Lec 33 - Matrix Vector Products

Lec 34 - Introduction to the Null Space of a Matrix

Lec 35 - Null Space 2: Calculating the null space of a matrix

Lec 36 - Null Space 3: Relation to Linear Independence

Lec 37 - Column Space of a Matrix

Lec 39 - Visualizing a Column Space as a Plane in R3

Lec 40 - Proof: Any subspace basis has same number of elements

Lec 41 - Dimension of the Null Space or Nullity

Lec 42 - Dimension of the Column Space or Rank

Lec 43 - Showing relation between basis cols and pivot cols

Lec 44 - Showing that the candidate basis does span C(A)

Lec 45 - A more formal understanding of functions

Lec 46 - Vector Transformations

Lec 47 - Linear Transformations

Lec 48 - Matrix Vector Products as Linear Transformations

Lec 49 - Linear Transformations as Matrix Vector Products

Lec 50 - Image of a subset under a transformation

Lec 51 - im(T): Image of a Transformation

Lec 52 - Preimage of a set

Lec 53 - Preimage and Kernel Example

Lec 54 - Sums and Scalar Multiples of Linear Transformations

Lec 55 - More on Matrix Addition and Scalar Multiplication

Lec 56 - Linear Transformation Examples: Scaling and Reflections

Lec 57 - Linear Transformation Examples: Rotations in R2

Lec 58 - Rotation in R3 around the X-axis

Lec 59 - Unit Vectors

Lec 60 - Introduction to Projections

Lec 61 - Expressing a Projection on to a line as a Matrix Vector prod

Lec 62 - Compositions of Linear Transformations 1

Lec 63 - Compositions of Linear Transformations 2

Lec 64 - Linear Algebra: Matrix Product Examples

Lec 65 - Matrix Product Associativity

Lec 66 - Distributive Property of Matrix Products

Lec 67 - Linear Algebra: Introduction to the inverse of a function

Lec 68 - Proof: Invertibility implies a unique solution to f(x)=y

Lec 69 - Surjective (onto) and Injective (one-to-one) functions

Lec 70 - Relating invertibility to being onto and one-to-one

Lec 71 - Determining whether a transformation is onto

Lec 72 - Linear Algebra: Exploring the solution set of Ax=b

Lec 73 - Linear Algebra: Matrix condition for one-to-one trans

Lec 74 - Linear Algebra: Simplifying conditions for invertibility

Lec 75 - Linear Algebra: Showing that Inverses are Linear

Lec 76 - Linear Algebra: Deriving a method for determining inverses

Lec 77 - Linear Algebra: Example of Finding Matrix Inverse

Lec 78 - Linear Algebra: Formula for 2x2 inverse

Lec 79 - Linear Algebra: 3x3 Determinant

Lec 80 - Linear Algebra: nxn Determinant

Lec 81 - Linear Algebra: Determinants along other rows/cols

Lec 82 - Linear Algebra: Rule of Sarrus of Determinants

Lec 83 - Linear Algebra: Determinant when row multiplied by scalar

Lec 84 - Linear Algebra: (correction) scalar muliplication of row

Lec 85 - Linear Algebra: Determinant when row is added

Lec 86 - Linear Algebra: Duplicate Row Determinant

Lec 87 - Linear Algebra: Determinant after row operations

Lec 88 - Linear Algebra: Upper Triangular Determinant

Lec 89 - Linear Algebra: Simpler 4x4 determinant

Lec 90 - Linear Algebra: Determinant and area of a parallelogram

Lec 91 - Linear Algebra: Determinant as Scaling Factor

Lec 92 - Linear Algebra: Transpose of a Matrix

Lec 93 - Linear Algebra: Determinant of Transpose

Lec 94 - Linear Algebra: Transpose of a Matrix Product

Lec 95 - Linear Algebra: Transposes of sums and inverses

Lec 96 - Linear Algebra: Transpose of a Vector

Lec 97 - Linear Algebra: Rowspace and Left Nullspace

Lec 98 - Lin Alg: Visualizations of Left Nullspace and Rowspace

Lec 99 - Linear Algebra: Orthogonal Complements

Lec 100 - Linear Algebra: Rank(A) = Rank(transpose of A)

Lec 101 - Linear Algebra: dim(V) + dim(orthogonoal complelent of V)=n