🎬AI Lectures40

This session introduces a brand-new course on building language models from scratch. You learn what language modeling is, where it’s used (speech recognition, translation, text generation, classification), and how different modeling families work. The class emphasizes implementing models yourself in Python and PyTorch, plus how to train and evaluate them.

GPUs (Graphics Processing Units) are critical for deep learning because they run thousands of simple math operations at the same time. Language models like Transformers rely on huge numbers of matrix multiplications, which are perfect for parallel processing. CPUs have a few strong cores for complex, step-by-step tasks, while GPUs have many simpler cores for doing lots of math in parallel. Using GPUs correctly can make training and inference dramatically faster.

This lesson explains what a vector really is from three connected views: a directed arrow in space, a coordinate like (4, 3), and a list of numbers like [4, 3]. Thinking of vectors as arrows makes direction and length feel natural, while coordinates make calculation easy. Both are the same thing described in different ways. You can move an arrow anywhere without changing the vector, as long as its direction and length stay the same.

This lesson explains Cramer's rule using geometry. A linear system like 2x + 5y = 7 and −3x + 4y = −1 can be written as Ax = v, where A’s columns are two vectors a1 and a2. The solution (x, y) tells how much to stretch and add a1 and a2 to land exactly on v.

The determinant is a single number attached to every square matrix that tells how a linear transformation scales area in 2D or volume in 3D. Its absolute value is the scale factor, and its sign tells whether the transformation keeps orientation or flips it. Think of dropping a 1-by-1 square (or 1-by-1-by-1 cube) into the transformation and measuring what size it becomes.

Three-dimensional linear transformations change the whole 3D space while keeping the origin fixed and all grid lines straight and parallel. This is just like in 2D, but now there are three axes: x, y, and z. These transformations stretch, rotate, shear (slant), or reflect the space without bending or curving it.

Dot product (also called inner product) takes two vectors and gives one number by multiplying matching coordinates and adding them. For example, with V=(3,2) and W=(4,-1), the dot product is 3*4 + 2*(-1) = 10. This single number is not just arithmetic; it measures how much V goes in the direction of W. If V points with W, the number is positive; if against, it’s negative; if perpendicular, it’s near zero.

A matrix with different numbers of rows and columns models a transformation between spaces of different sizes. For example, a 3-by-2 matrix takes 2D vectors from the flat plane and turns them into 3D vectors in space. The columns of the matrix tell you exactly where the basic 2D directions (i-hat and j-hat) end up in 3D. Using this rule, any 2D input can be mapped by combining those columns.

Matrices represent linear transformations, which are rules that stretch, rotate, shear, or squash space while keeping straight lines straight and the origin fixed. When you multiply matrices, you are chaining these transformations: first do one change to space, then do the next. Some transformations lose information by collapsing dimensions, like flattening a whole plane onto a line, and those cannot be undone.

This lesson builds an intuitive, picture-first understanding of eigenvalues and eigenvectors. Instead of starting with heavy equations, it treats a matrix as a machine that reshapes the whole 2D plane and then looks for special directions that do not turn. These special directions are eigenvectors, and the stretch or shrink amount along them is the eigenvalue. You will see why some vectors change both length and direction, while a few special ones only change length.