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Linear Algebra

Vectors

  • Ordered list of numbers to represent points in space, directions or any other quantities that have magnitude and direction
  • Addition of vectors is applied element wise and produces a vector of the same shape
  • Scalar multiplication is applied element wise and produces a vector
  • Dot product is sum product of the vectors and produces a scalar
  • Cross product is the multiplication of direction and magnitude and produces a vector perpendicular to both
    • you have to multiply all coordinates and directions

Matrices

  • 2D vectors
  • Represent a system of linear equations, data tables or transformations
  • Addition is applied element wise and produces a matrix of the same shape
  • Scalar multiplication is applied element wise and you get a matrix of the same shape
  • Matrix multiplication is applied vector wise and you get a matrix with a number of rows same as the first matrix and a number of columns the same as the second
  • Transpose is to flip along the diagonal
  • Determinant is a scalar value and a property of a square matrix
    • For a 2x2 square matrix of (abcd) it is calculated like ad−bcad - bc or
    • For a 3x3 square matrix of (abcdefghi) it is calculated like a×det(efhi)−b×det(dfgi)+c×det(degh)a \times det(efhi) - b \times det(dfgi) + c \times det(degh)
  • Inverse matrix is one that AA−1=IAA^-1=I
    • The square matrix is invertible IFF the determinant is non-zero