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1. Scalar, Vector, and Matrix

1) Scalar

• a single number
• $s \in \mathbb{R}$

2) Vector

• an ordered list of numbers
• $\mathbf{x} = \begin{bmatrix}x_1 \\x_2 \\\vdots \\x_n\end{bmatrix}\in\mathbb{R^n}$

3) Matrix

• a two-dimensional array of numbers
• $A = \begin{bmatrix} 1 & 6 \\ 3 & 4 \\ 5 & 2 \\ \end{bmatrix}\in\mathbb{R}^{3 \times 2}$
• Row vector : a horizontal vector
• Column vector : a vertical vector

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2. Column Vector and Row vector

1) Column Vector

• A vector of n-dimension → a column vector
• $\mathbf{x} = \begin{bmatrix}x_1 \\x_2 \\\vdots \\x_n\end{bmatrix}\in\mathbb{R^n}=\mathbb{R}^{n \times 1}$

2) Row Vector

• transpose

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

• Example (transpose of a matrix)

  $A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}, \quad A^{\mathsf{T}} = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}$

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3. Matrix Notations

1) Square matrix

• $A\in\mathbb{R}^{n \times n}$

2) Rectangular matrix

• $A\in\mathbb{R}^{m \times n}$

3) Transpose of matrix

• $\mathbf{A}^{\mathsf{T}}$

4) $A_{ij}$

• $(i, j)$ -th component of $A$
• $A_{i, :}$ : $i$ - th row vector of $A$
• $A_{:, i}$ : $i$ - th column vector of $A$

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4. Vector/Matrix Additions and Multiplications

1) $C = A + B$

• $C_{ij} = A_{ij} + B_{ij}$
• $A, B, C$ should have the same size

2) $ca, cA$

• Scalar multiple of vector / matrix
  $2 \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix} = \begin{bmatrix} 6 \\ 4 \\ 2 \end{bmatrix}$

3) $C = AB$

• Matrix - matrix multiplication

• $C_{ij} = \sum_{k} A_{i,k} B_{k,j}$

• $A \in \mathbb{R}^{m \times n},\ B \in \mathbb{R}^{n \times p} \Rightarrow AB \in \mathbb{R}^{m \times p}$

• $\begin{bmatrix}1 & 2 & 3 \\4 & 5 & 6\end{bmatrix}\begin{bmatrix}1 & 0 \\0 & 1 \\1 & -1\end{bmatrix}=\begin{bmatrix}4 & -1 \\10 & -1\end{bmatrix}$

  ▶️ $4=(1\times1)+(2\times0)+(3\times1)$

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5. Matrix multiplication is NOT commutative

• $AB \neq BA$