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Linear algebra (bridge)

The minimum of linear algebra that multivariable calculus needs: vectors, the dot product and matrices. The full story lives in the linear algebra chapter of Math of AI.

3 topics

This portal is about calculus, but from the gradient onwards calculus speaks the language of linear algebra: derivatives of functions of many variables are vectors and matrices. These three topics are the bridge. For eigenvalues, the SVD and embeddings, see Math of AI · Linear algebra.

Topics

Vectors

Lists of numbers $v = (v_1, \dots, v_n) \in \R^n$ that can be added and scaled. Geometrically, arrows with a length and a direction; computationally, arrays.

Fundamental

Dot product

$u \cdot v = \sum_i u_i v_i = \norm u \norm v \cos\theta$ . It measures how much two vectors point the same way, and it is the single most executed operation in machine learning.

Fundamental

Matrices and linear maps

A matrix $A \in \R^{m \times n}$ is a linear map $x \mapsto Ax$ from $\R^n$ to $\R^m$ . The derivative of a function of several variables is a matrix (the Jacobian), and a layer of a neural network is $Wx + b$ .

University

Where this area leads in computing

Topics

Vectors

Dot product

Matrices and linear maps

Where this area leads in computing

ℒ AI and machine learning ★★★★★

3D Computer graphics ★★★★★

⚛ Physics and simulation ★★★★★

⚙ Robotics and control ★★★★★

⇄ Optimization and systems ★★★★★