1. Scalar, Vector, and Matrix

• Scalar : a single number

  $$ s \in \mathbb{R} (lower case), \quad e.g.,\;3.8 $$

• Vector : an ordered list of numbers,

$$ e.g.\quad\mathbf{x} = \begin{bmatrix}x_1\\x_2\\\vdots\\x_n\end{bmatrix} \in \mathbb{R}^n (boldface, lower-case) \quad e.g., \mathbf{x} = \begin{bmatrix} 1\\0\\2 \end{bmatrix} \in \mathbb{R}^3 $$

• Matrix : a two-dimensional array of numbers

$$ e.g., A = \begin{bmatrix} 1&6\\3&4\\5&2 \end{bmatrix} \in \mathbb{R}^{3 \times 2} (capital\;letter) $$

2. Column Vector and Row Vector

※ Column in Vertical Vector (Don’t be Confused!)

• A vector of $n$-dimension is usually a column vector, i.e., a matrix of the size $n \times 1$

$$ \mathbf{x} = \begin{bmatrix} x_1\\x_2\\ \vdots \\x_n \end{bmatrix} \in \mathbb{R}^n = R^{n \times 1} $$

• Thus, a row vector is usually written as its transpose, i.e.,

$$ \mathbf{x}^T = \begin{bmatrix} x_1\\x_2\\ \vdots \\x_n \end{bmatrix}^T = \begin{bmatrix} x_1 & x_2 & \cdots & x_n \end{bmatrix} \in \mathbb{R}^{1 \times n} $$

3. Matrix Notations

• Square matrix (#rows $=$ #columns)

$$ A \in \mathbb{R}^{n \times n}\quad e.g., \; \mathbf{B} = \begin{bmatrix} 1&6\\2&4\end{bmatrix} $$

• Rectangular matrix (possible: #rows $\neq$ #columns)