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Lec 43 - Showing relation between basis cols and pivot cols

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

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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 38 - Null Space and Column Space Basis

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 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