Linear Algebra

Core Theoretical Principles

Linear algebra provides the mathematical language of multi-dimensional spaces, computer graphics, and machine learning.

1. The Geometric Meaning of Determinants

The determinant of an $n \times n$ matrix $A$ measures the factor by which the linear map $T(\mathbf{x}) = A\mathbf{x}$ scales $n$-dimensional volumes:

$$ \operatorname{Vol}(T(S)) = |\det(A)| \cdot \operatorname{Vol}(S) $$

If $\det(A) = 0$, the transformation collapses space into a lower dimension, implying non-invertibility ($\ker(A) \neq \{\mathbf{0}\}$).

2. Sarrus’ Rule for $3 \times 3$ Determinants

For a general $3 \times 3$ matrix $A$:

$$ A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} $$

By copying the first two columns to the right of the matrix, the determinant equals the sum of the products of the three downward diagonals minus the sum of the products of the three upward diagonals:

$$ \det(A) = (a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32}) - (a_{13}a_{22}a_{31} + a_{11}a_{23}a_{32} + a_{12}a_{21}a_{33}) $$

Explore our full video lecture below for animated demonstrations of this visual mnemonic.