TBD Will connect to backpropagation and gradient descent. (Gemini)
1. Dot Products: Measuring "Similarity"
Think of a dot product as a way to ask a computer: "How much do these two things agree?"
In ML, we represent features (like the words in a sentence or pixels in an image) as vectors (lists of numbers). When we take the dot product of two vectors, we get a single number that tells us how aligned they are.
Positive Result: The vectors are "pointing" in the same direction (they are similar).
Zero Result: They are perpendicular (they have nothing in common).
Negative Result: They are opposites.
Example: In a recommendation system, one vector might represent "Your Interests" and another "Movie Genres." A high dot product suggests a strong match between your tastes and that movie.
2. Matrices: Handling Data at Scale
If a dot product is a single comparison, a matrix is a massive filing cabinet that lets you perform thousands of those comparisons simultaneously.
Efficiency (Vectorization): Instead of writing a "for-loop" to calculate the weighted sum for every single neuron in a neural network, we pack all the inputs into one matrix and all the weights into another. One single matrix multiplication can process an entire batch of data at once.
Space Transformation: Matrices act as "instructions" that rotate, stretch, or compress data. This allows a model to transform raw, messy input data into a clean mathematical space where patterns are easier to spot.
3. Hardware Optimization (The GPU Factor)
The reason AI has exploded recently isn't just better math, it's better hardware.
Computer processors (CPUs) are built to handle complex tasks one by one. However, Graphics Processing Units (GPUs) are designed to do thousands of simple tasks in parallel.
Since matrix multiplication is just a huge collection of simple additions and multiplications, GPUs can perform these operations thousands of times faster than a standard processor.