[Linear Algebra] Vector Matrix Notation

Jason Lee·2022년 8월 4일
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Linear Algebra

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Scalar, Vector and Matrix

  • Scalar : sRs \in \mathbb{R} (single number)
  • Vector : xRn\textbf{x} \in \mathbb{R}^n (list of numbers)
  • Matrix : ARm×nA \in \mathbb{R}^{m \times n} (two-dimensional array of numbers)

Column Vector and Row Vector

  • Column vector : x=[x1x2xn]Rn×1\textbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} \in \mathbb{R}^{n \times 1}

  • Row vector : xT=[x1x2xn]T=[x1x2xn]R1×n\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 : ARn×nA \in \mathbb{R}^{n \times n}
  • Rectangular matrix : ARm×nA \in \mathbb{R}^{m \times n}
  • Transpose of matrix : ATRn×mA^T \in \mathbb{R}^{n \times m} (mirroring across the main diagonal)
  • (i, j)-th component of AA : AijA_{ij}
  • i-th row vector of AA : Ai,:A_{i,:}
  • j-th column vector of AA : A:,jA_{:,j}
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