Pranav

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CONCEPTUAL MAP

NOTE: A~B $\implies$ A with respect to B

V: Vectors
- V.base: General definition of vector and specific interpretations
  - Physics context (relative change in position)
  - Computer science and data science context (ordered list of values)
  - Mathematics, i.e. generalised context (generalises all other contexts)
  - Relation between geometric and numeric interpretations
- V.Op: Vector operations and their meaning w.r.t. geometric interpretation
  NOTE: Full clarity of some topics requires understanding linear transformations
  - Vector addition (obtaining the resultant vector of a series of vector movements)
  - Scalar product (scaling a vector, i.e. changing its magnitude while maintaining direction)
  - Vector product
  - Dot product and its meaning (discussed later)
  - Cross product and its meaning (discussed later)
- V.Sp: Span of a set of vectors
- V.M: Matrices as vectors of vectors
- V.VS: Vector spaces
  - V.VS.Dim: Dimensions of a vector space (in every interpretation, i.e. every context)
  - V.VS.B: Basis of a vector space
    KEY POINT: Every vector of the vector space is a linear combination of the basis vectors
  - LT~V.VS: Linear transformations as transformations of vector spaces
    - Representing a LT as a transformation of basis vectors
      KEY POINT: Every transformedd vector is a linear combination of the transformed basis vectors
    - Reducing the dimensionality of a vector space through LT

Matrices are deeply tied to linear transformations, so always consider matrix-related topics in the context of LT…

M: Matrices
… extends from V.M
- M.base: General definition of matrix and specific interpretations
- M.Op: Matrix operations and their meaning w.r.t. vectors
  - Matrix addition (ordered sequence of multiple vector additions)
  - Scalar product (ordered sequence of the scaling of multiple vectors)
  - Matrix multiplication (ordered sequence of linear combinations of multiple vectors)
- LT~M: Linear transformations as matrices
  - Representing a LT as a matrix
  - Linearly transforming a vector with matrix multiplication
  - Relating composition of functions to composition of LT’s to composition of matrices
    Note that LT is a kind of function
- M.Det: Determinant
  - M.Det~LT: Determinant as a number related to a LT
    More specifically, it gives the scale and orientation of the LT
    NOTE: Scaling $\implies$ Scaling of any subspace of the original space
    - Validating this representation of scale and orientation
    - What does zero determinant mean?
  - Calculation and justification of the calculation method
  - Key results on determinants and their validation
    - $\det(A B) = \det(A) \det(B)$
- M.Inv: Inverse of a matrix
  - M.Det~LT: Inverse of a matrix as inverse (i.e. reversal) of a LT
    LT is a function; think in terms of inverse of functions
  - Relation between inverse and identity transformations
  - Relation between determinant and inverse
- M.SoE: Systems of equations as matrices
  - M.SoE~LT: SoE as finding the preimage of a vector w.r.t. a LT
    Looking for the vector(s) that is mapped by the LT to the given vector
  - Zero determinant and possible solutions given coefficient
    Connect with LT
  - Specifying zero determinant cases using rank
    Connect with LT
  - Obtaining the solution of a SoE using the inverse of the coefficient matrix

Generalising linear transformation topics…

LT: Linear transformation (general concepts)
NOTE: We shall take for granted that LT’s are represented as matrices
- REFERENCE FOR INTUITION: Inverse matrices, column space and null space | Chapter 7, Essence of linear algebra
- LT.base: Basics
  - LT as a function defined on a vector space
  - Representing LT as a transformation of the basis vectors
  - Representing LT as a matrix (directly related to the above)
- LT.Rank: Rank of a matrix a.k.a. rank of a LT
- LT.CS: Column space of a matrix
  Essentially, the span of the transformed basis vectors, i.e. the columns
  - LT.Rank~LT.CS: Rank is the number of dimensions in the column space
  - “Full rank” matrix $\implies$ Rank equals the number of columns $\implies$ Maximum dimensions for the column space
- LT.DimRed: Dimensionality reduction of vector space with LT
- LT.NS: Nullspace of a matrix a.k.a. nullspace of a LT
  … related to LT.DimRed
  - The set of vectors that get mapped by the LT to a null vector, i.e. the origin
  - LT.NS~M.SoE: The set of all solution to a homogenous system of equations represented by the matrix