Linear algebra

Categories: Algebra, Mathematics, Sciences
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Course Content

Vectors and spaces

  • Vectors
  • Vector intro for linear algebra
  • Real coordinate spaces
  • Adding vectors algebraically & graphically
  • Multiplying a vector by a scalar
  • Vector examples
  • Unit vectors intro
  • Parametric representations of lines
  • Linear combinations and spans
  • Linear combinations and span
  • Linear dependence and independence
  • Introduction to linear independence
  • More on linear independence
  • Span and linear independence example
  • Subspaces and the basis for a subspace
  • Linear subspaces
  • Basis of a subspace
  • Vector dot and cross products
  • Vector dot product and vector length
  • Proving vector dot product properties
  • Proof of the Cauchy-Schwarz inequality
  • Vector triangle inequality
  • Defining the angle between vectors
  • Defining a plane in R3 with a point and normal vector
  • Cross product introduction
  • Proof: Relationship between cross product and sin of angle
  • Dot and cross product comparison/intuition
  • Vector triple product expansion (very optional)
  • Normal vector from plane equation
  • Point distance to plane
  • Distance between planes
  • Matrices for solving systems by elimination
  • Solving a system of 3 equations and 4 variables using matrix row-echelon form
  • Solving linear systems with matrices
  • Using matrix row-echelon form in order to show a linear system has no solutions
  • Null space and column space
  • Matrix vector products
  • Introduction to the null space of a matrix
  • Null space 2: Calculating the null space of a matrix
  • Null space 3: Relation to linear independence
  • Column space of a matrix
  • Null space and column space basis
  • Visualizing a column space as a plane in R3
  • Proof: Any subspace basis has same number of elements
  • Dimension of the null space or nullity
  • Dimension of the column space or rank
  • Showing relation between basis cols and pivot cols
  • Showing that the candidate basis does span C(A)

Matrix transformations

Alternate coordinate systems (bases)

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