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Lecture 40: Proof: Any subspace basis has same number of elements

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Proof: Any subspace basis has same number of elements

Course Description :

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.

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  1. Introduction to matrices
  2. Matrix multiplication (part 1)
  3. Matrix multiplication (part 2)
  4. Inverse Matrix (part 1)
  5. Inverting matrices (part 2)
  6. Inverting Matrices (part 3)
  7. Matrices to solve a system of equations
  8. Matrices to solve a vector combination problem
  9. Singular Matrices
  10. 3-variable linear equations (part 1)
  11. Solving 3 Equations with 3 Unknowns
  12. Linear Algebra: Introduction to Vectors
  13. Linear Algebra: Vector Examples
  14. Linear Algebra: Parametric Representations of Lines
  15. Linear Combinations and Span
  16. Linear Algebra: Introduction to Linear Independence
  17. More on linear independence
  18. Span and Linear Independence Example
  19. Linear Subspaces
  20. Linear Algebra: Basis of a Subspace
  21. Vector Dot Product and Vector Length
  22. Proving Vector Dot Product Properties
  23. Proof of the Cauchy-Schwarz Inequality
  24. Linear Algebra: Vector Triangle Inequality
  25. Defining the angle between vectors
  26. Defining a plane in R3 with a point and normal vector
  27. Linear Algebra: Cross Product Introduction
  28. Proof: Relationship between cross product and sin of angle
  29. Dot and Cross Product Comparison/Intuition
  30. Matrices: Reduced Row Echelon Form 1
  31. Matrices: Reduced Row Echelon Form 2
  32. Matrices: Reduced Row Echelon Form 3
  33. Matrix Vector Products
  34. Introduction to the Null Space of a Matrix
  35. Null Space 2: Calculating the null space of a matrix
  36. Null Space 3: Relation to Linear Independence
  37. Column Space of a Matrix
  38. Null Space and Column Space Basis
  39. Visualizing a Column Space as a Plane in R3
  40. Proof: Any subspace basis has same number of elements
  41. Dimension of the Null Space or Nullity
  42. Dimension of the Column Space or Rank
  43. Showing relation between basis cols and pivot cols
  44. Showing that the candidate basis does span C(A)
  45. A more formal understanding of functions
  46. Vector Transformations
  47. Linear Transformations
  48. Matrix Vector Products as Linear Transformations
  49. Linear Transformations as Matrix Vector Products
  50. Image of a subset under a transformation