Scalars, Vectors, Matrices and Tensors

• Scalars
  • Just a single number.
  • ex) $s \in \mathbb{R}$, $n \in \mathbb{N}$
• Vectors
  • An array of numbers.
  • ex) $\bm{x} \in \mathbb{R}^n$
• Matrices
  • A 2-D array of numbers.
  • ex) $\bm{A} \in \mathbb{R}^{m\times n}$
• Tensors
  • An array with more than two axes.

---

• Transpose \(\left( \bm{A}^\top \right)_{i,j} = \bm{A}_{j,i} \\ \left( \bm{A} \bm{B} \right) ^\top = \bm{B}^\top \bm{A}^\top\)

  • A transpose of a column vector is a row vector, and vice versa.
  • A transpose of a scalar is itself.