Linear algebra is the branch of mathematics concerned with vector spaces, linear transformations, and systems of linear equations. It provides the foundational language for countless fields—from solving differential equations and optimizing machine learning models to rendering computer graphics and analyzing networks. At its core, linear algebra studies how linear combinations of vectors span spaces, how matrices encode transformations, and how eigenvalues reveal fundamental structures. Understanding that matrices represent linear maps between vector spaces—not just grids of numbers—unlocks the deeper insights that make this subject so powerful across science and engineering.
What This Cheat Sheet Covers
This topic spans 22 focused tables and 185 indexed concepts. Below is a complete table-by-table outline of this topic, spanning foundational concepts through advanced details.
Table 1: Vector Operations
| Operation | Example | Description |
|---|---|---|
\begin{bmatrix}1\\2\end{bmatrix} + \begin{bmatrix}3\\4\end{bmatrix} = \begin{bmatrix}4\\6\end{bmatrix} | • Component-wise sum of two vectors of equal dimension • forms the parallelogram rule geometrically. | |
3 \begin{bmatrix}1\\2\end{bmatrix} = \begin{bmatrix}3\\6\end{bmatrix} | • Multiplies each component by a scalar • scales the vector's magnitude without changing direction (unless scalar is negative). | |
\begin{bmatrix}1\\2\\3\end{bmatrix} \cdot \begin{bmatrix}4\\5\\6\end{bmatrix} = 32 | • Sum of component-wise products yielding a scalar • measures projection and angle via \mathbf{u} \cdot \mathbf{v} = \lvert \mathbf{u}\rvert\lvert \mathbf{v}\rvert\cos\theta. |