Vector

Definition:

• Collection of number in 1 column
• Can represent:
  • A point in space in $n$-dimension
  • A direction
  • A linear function
  • Time series
• Has notation $\left[\begin{array}{c}a\\ b\\ c\end{array}\right]$ or $(a,b,c)$
• Can be stacked up

Vector components:

• $x(i)$ denotes the $i$-th component of vector $x$

Vector Transformation:

• Transpose, ${}^T$:
  • Transform column-vector $x$ to row format
  • $x^T=[x_1{x}_2...x_n]$
  • $x^{TT}=x$

Scalar Product

Unit Vector

Ones Vector

Euclidean Length

• $||x||$ denotes the length of the vector

Vector’s Variance:

• Variance between the elements of the vector
  • as wel as std

De-meaned Vector

Orthonormal Vector:

• Orthogonality
• Projecting a point on the span of orthonormal vectors can be done by projecting on each vector independently.

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