Linear Algebra

Linear Algebra

Dr. Trefor Bazett via YouTube Direct link

Are Matrix Transformations and Linear Transformation the same? Part I

26 of 82

26 of 82

Are Matrix Transformations and Linear Transformation the same? Part I

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Classroom Contents

Linear Algebra

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  1. 1 What's the big idea of Linear Algebra? **Course Intro**
  2. 2 What is a Solution to a Linear System? **Intro**
  3. 3 Visualizing Solutions to Linear Systems - - 2D & 3D Cases Geometrically
  4. 4 Rewriting a Linear System using Matrix Notation
  5. 5 Using Elementary Row Operations to Solve Systems of Linear Equations
  6. 6 Using Elementary Row Operations to simplify a linear system
  7. 7 Examples with 0, 1, and infinitely many solutions to linear systems
  8. 8 Row Echelon Form and Reduced Row Echelon Form
  9. 9 Back Substitution with infinitely many solutions
  10. 10 What is a vector? Visualizing Vector Addition & Scalar Multiplication
  11. 11 Introducing Linear Combinations & Span
  12. 12 How to determine if one vector is in the span of other vectors?
  13. 13 Matrix-Vector Multiplication and the equation Ax=b
  14. 14 Matrix-Vector Multiplication Example
  15. 15 Proving Algebraic Rules in Linear Algebra --- Ex: A(b+c) = Ab +Ac
  16. 16 The Big Theorem, Part I
  17. 17 Writing solutions to Ax=b in vector form
  18. 18 Geometric View on Solutions to Ax=b and Ax=0.
  19. 19 Three nice properties of homogeneous systems of linear equations
  20. 20 Linear Dependence and Independence - Geometrically
  21. 21 Determining Linear Independence vs Linear Dependence
  22. 22 Making a Math Concept Map | Ex: Linear Independence
  23. 23 Transformations and Matrix Transformations
  24. 24 Three examples of Matrix Transformations
  25. 25 Linear Transformations
  26. 26 Are Matrix Transformations and Linear Transformation the same? Part I
  27. 27 Every vector is a linear combination of the same n simple vectors!
  28. 28 Matrix Transformations are the same thing as Linear Transformations
  29. 29 Finding the Matrix of a Linear Transformation
  30. 30 One-to-one, Onto, and the Big Theorem Part II
  31. 31 The motivation and definition of Matrix Multiplication
  32. 32 Computing matrix multiplication
  33. 33 Visualizing Composition of Linear Transformations **aka Matrix Multiplication**
  34. 34 Elementary Matrices
  35. 35 You can "invert" matrices to solve equations...sometimes!
  36. 36 Finding inverses to 2x2 matrices is easy!
  37. 37 Find the Inverse of a Matrix
  38. 38 When does a matrix fail to be invertible? Also more "Big Theorem".
  39. 39 Visualizing Invertible Transformations (plus why we need one-to-one)
  40. 40 Invertible Matrices correspond with Invertible Transformations **proof**
  41. 41 Determinants - a "quick" computation to tell if a matrix is invertible
  42. 42 Determinants can be computed along any row or column - choose the easiest!
  43. 43 Vector Spaces | Definition & Examples
  44. 44 The Vector Space of Polynomials: Span, Linear Independence, and Basis
  45. 45 Subspaces are the Natural Subsets of Linear Algebra | Definition + First Examples
  46. 46 The Span is a Subspace | Proof + Visualization
  47. 47 The Null Space & Column Space of a Matrix | Algebraically & Geometrically
  48. 48 The Basis of a Subspace
  49. 49 Finding a Basis for the Nullspace or Column space of a matrix A
  50. 50 Finding a basis for Col(A) when A is not in REF form.
  51. 51 Coordinate Systems From Non-Standard Bases | Definitions + Visualization
  52. 52 Writing Vectors in a New Coordinate System **Example**
  53. 53 What Exactly are Grid Lines in Coordinate Systems?
  54. 54 The Dimension of a Subspace | Definition + First Examples
  55. 55 Computing Dimension of Null Space & Column Space
  56. 56 The Dimension Theorem | Dim(Null(A)) + Dim(Col(A)) = n | Also, Rank!
  57. 57 Changing Between Two Bases | Derivation + Example
  58. 58 Visualizing Change Of Basis Dynamically
  59. 59 Example: Writing a vector in a new basis
  60. 60 What eigenvalues and eigenvectors mean geometrically
  61. 61 Using determinants to compute eigenvalues & eigenvectors
  62. 62 Example: Computing Eigenvalues and Eigenvectors
  63. 63 A range of possibilities for eigenvalues and eigenvectors
  64. 64 Diagonal Matrices are Freaking Awesome
  65. 65 How the Diagonalization Process Works
  66. 66 Compute large powers of a matrix via diagonalization
  67. 67 Full Example: Diagonalizing a Matrix
  68. 68 COMPLEX Eigenvalues, Eigenvectors & Diagonalization **full example**
  69. 69 Visualizing Diagonalization & Eigenbases
  70. 70 Similar matrices have similar properties
  71. 71 The Similarity Relationship Represents a Change of Basis
  72. 72 Dot Products and Length
  73. 73 Distance, Angles, Orthogonality and Pythagoras for vectors
  74. 74 Orthogonal bases are easy to work with!
  75. 75 Orthogonal Decomposition Theorem Part 1: Defining the Orthogonal Complement
  76. 76 The geometric view on orthogonal projections
  77. 77 Orthogonal Decomposition Theorem Part II
  78. 78 Proving that orthogonal projections are a form of minimization
  79. 79 Using Gram-Schmidt to orthogonalize a basis
  80. 80 Full example: using Gram-Schmidt
  81. 81 Least Squares Approximations
  82. 82 Reducing the Least Squares Approximation to solving a system

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