[Linear Algebra] Vector Matrix Notation

Scalar, Vector and Matrix

• Scalar : $s \in \mathbb{R}$ (single number)
• Vector : $\textbf{x} \in \mathbb{R}^n$ (list of numbers)
• Matrix : $A \in \mathbb{R}^{m \times n}$ (two-dimensional array of numbers)

Column Vector and Row Vector

• Column vector : $\textbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} \in \mathbb{R}^{n \times 1}$

• Row vector : $\textbf{x}^T = {\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}}^T = \begin{bmatrix} x_1 & x_2 & \cdots & x_n \\ \end{bmatrix} \in \mathbb{R}^{1 \times n}$

Matrix Notations

• Square matrix : $A \in \mathbb{R}^{n \times n}$
• Rectangular matrix : $A \in \mathbb{R}^{m \times n}$
• Transpose of matrix : $A^T \in \mathbb{R}^{n \times m}$ (mirroring across the main diagonal)
• (i, j)-th component of $A$ : $A_{ij}$
• i-th row vector of $A$ : $A_{i,:}$
• j-th column vector of $A$ : $A_{:,j}$