Day-1 (Linear Algebra — Vectors, Matrices)
A simple introduction to the essence of linear algebra and its concepts.
What I learned today
- Why this field exists, and what its intuition is in this world
- What vectors are, and how their operations work, both mathematically and visually
- What matrices are and why we need them
- What span, basis, linear transformation, 3D transformation, and determinant are
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How it works
- In physics, a vector is an arrow that starts at the origin (0) and has a specific length and direction (e.g. see these diagrams)
- In computer science, a vector is an ordered list of numbers (e.g. [5,9], [-2,0,3])
- A matrix is just a combination of such vectors arranged in an n-dimensional grid, whereas a vector has only 1 dimension
- Like numeric addition, subtraction, and multiplication, we can perform similar operations on vectors and matrices
- A determinant is just the area or volume under the vectors
What confused me
- At first, I did not understand why matrix multiplication works this way. Later I understood that one matrix holds the data or vectors, and the other describes how the space transforms. Multiplying them gives the transformed matrix or vectors.
- Another (smaller) confusion was what a determinant is and how it works. Later I learned that it is just the area or volume under the vectors, and it gets a negative sign if the space is flipped along an axis.
Why it matters
- It matters to me because it is heavily used in computer science, especially in AI and machine learning.
- It matters to me because it is used in image processing algorithms and geometry processing.
- In 3D and AR/VR, it is widely used for space understanding, object creation, interaction, etc.
Tomorrow
- I will take a deeper look at column space, types of matrices, dot products etc.
Source
- YouTube: 3Blue1Brown — Essence of Linear Algebra
