Course notes: Essence of linear algebra (3b1b)

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The Youtube playlist Essence of linear algebra is a famous short course for linear algebra. It was super helpful for my previous interview for admission of the master of data science programme in HKU (which I got a offer on 2019 but forfeited eventually). Now I wanted to revisit this course again to solidly my understand on these concepts, in preparation of future study on deep learning models. I will take notes of the course here.

Vectors | Chapter 1

Imagine vector addition and scalar multiplication in a visual way (in a coordinate system).

Linear combinations, span, and basis vectors | Chapter 2

  • Linear combination: a vector that can be expressed as a linear combination of other vectors.
  • Span: the set of all possible linear combinations of a set of vectors.
  • Basis: a set of vectors that are linearly independent and span the vector space.

Linear transformations and matrices | Chapter 3

  • Linear transformation: a function that maps vectors to vectors and preserves the linear combination (lines remain lines, origin remains fixed).
  • Linear transformation on a given vector can be represented by a matrix multiplication, e.g. a 2D linear transformation can be represented by a 2x2 matrix multiplication on the vector (x, y), the original basis vectors (1, 0) and (0, 1) are transformed to the new basis vectors (a, c) and (b, d) : \(\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} \begin{bmatrix} x \\ y \\ \end{bmatrix}\)
  • Every time you see a matrix, you can interpret it as a certain transformation of the space.

Matrix multiplication as composition | Chapter 4

  • Matrix multiplication is a composition of linear transformations, e.g., a 2x2 matrix multiplication can be interpreted as a composition of two linear transformations (e.g., rotation and shear), each of which is a 2D linear transformation represented by a 2x2 matrix multiplication.
  • Matrix multiplication does not have the commutative property. This means that for matrices A and B: $A * B ≠ B * A$
  • Matrix multiplication does have associativity, i.e., for matrices A, B, and C: $(A * B) * C = A * (B * C)$