② Linear algebra
§1. Vectors and matrices¶
Vectors¶
\[\begin{equation*}
x = (x_1, x_2, \ldots, x_n)
\end{equation*}\]
Matrices¶
\[\begin{equation*}
A = \begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1k} \\
a_{21} & a_{22} & \cdots & a_{2k} \\
\vdots & \vdots & & \vdots \\
a_{n1} & a_{11} & \cdots & a_{nk} \\
\end{bmatrix}
\end{equation*}\]
\[\begin{equation}
(AB)' = B'A'
\end{equation}\]
§2. Systems of equations¶
Gaussian elimination¶
§3. Vector spaces¶
Vector spaces¶
TBD
A vector space over a field \(\mathbb{F}\) is a set \(V\) along with two operations:
- \(+\colon V \times V \to V\) (vector addition)
- \(\cdot\colon \mathbb{F} \times V \to V\) (scalar multiplication)
that satisfies the following properties:
- TBD
We call the elements of \(V\) vectors and the elements of \(\mathbb{F}\) scalars.
Dimensions¶
TBD
Bases¶
TBD
§4. Ranks, inverses, and determinants¶
TBD
Orthogonality¶
§5. Eigenvalues and eigenvectors¶
Perron-Frobenius theorem¶
TBD
§6. Factorizations¶
Singular value decomposition¶
\[\begin{equation}
A = U \Sigma V^\top
\end{equation}\]
Jordan decomposition¶
Schur decomposition¶
Cholesky decomposition¶
§7. Definite matrices¶
§8. Matrix inequalities¶
TBD
§9. Matrix stacking¶
Kronecker product¶
TBD
Vec-operator¶
TBD
Resources¶
Computational implementation¶
Julia
MATLAB
Textbooks¶
Visual connections¶
References¶
TBD