6  Vectors

Aims

Learning outcomes

A large number of statistical models use vectors and matrices, both for compact representations, and for the calculations, e.g. parameter estimates.

6.1 Vectors

  • A vector is an ordered set of number
  • These numbers, e.g. in vector \(\mathbf{x}\) can be expressed as a row \(\mathbf{x}=[6\quad 0\quad 5 \dots1]\)
  • or as a column \(\mathbf{x}=\begin{bmatrix} 6 \\ 0 \\ 5 \\ \vdots \\ 1 \end{bmatrix}\)
  • the number of elements in a vector is referred to as its dimension and we often use \(n\) to express \(n\)-dimensional vector, where \(n\) can be any natural number
  • here, we denote vectors using small bold font \(\mathbf{x}\), other notations may include an arrow \(\vec x\) or overline \(\overline{x}\)
  • also parentheses are used interchangeably with square bracket, e.g. \(\mathbf{x}=[6\quad 0\quad 5 \dots1]\) can be written as \(\mathbf{x}=(6\quad 0\quad 5 \dots1)\) or \(\begin{pmatrix} 6\\ 0\\ 5\\ \vdots \\ 1 \end{pmatrix}\)

6.2 Operations on vectors

Given two vectors of the same dimension: \(\mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ \vdots \\ x_n \end{bmatrix}\) and \(\mathbf{y}=\begin{bmatrix} y_1 \\ y_2 \\ y_3 \\ \vdots \\ y_n \end{bmatrix}\)

Addition: we add two vectors, element by element \(\mathbf{x} + \mathbf{y}=\begin{bmatrix} x_1 + y_1 \\ x_2 + y_2 \\ x_3 + y_3 \\ \vdots \\ x_n + y_n \end{bmatrix}\)

Scalar multiplication: we can multiply vector by a numerical value, scalar, denoted as \(\gamma\): \[\gamma \cdot \mathbf{x} =\begin{bmatrix} \gamma \cdot x_1 \\ \gamma \cdot x_2 \\ \gamma \cdot x_3 \\ \vdots \\ \gamma \cdot x_n \end{bmatrix}\]

Difference \(\mathbf{x} - \mathbf{y}\) can be written as \(\mathbf{x} + (-1) \cdot \mathbf{y}\), thus we multiply second vector with \(-1\) and then add two vectors

Linear combination of vectors: the vector \(\gamma \cdot \mathbf{x} + \delta \cdot \mathbf{y}\) is said to be a linear combination of \(\mathbf{x}\) and \(\mathbf{y}\): \[\gamma \cdot \mathbf{x} + \delta \cdot \mathbf{y} =\begin{bmatrix} \gamma \cdot x_1 + \delta \cdot y_1 \\ \gamma \cdot x_2 + \delta \cdot y_2\\ \gamma \cdot x_3 + \delta \cdot y_3\\ \vdots \\ \gamma \cdot x_n + \delta \cdot y_n \end{bmatrix}\]

Inner product of vectors is given by: \[\mathbf{x} \cdot \mathbf{y} = x_1 \cdot y_1 + x_2 \cdot y_2 + \dots x_n \cdot y_n = \displaystyle\sum_{i=1}^{n}x_i\cdot y_i\]

Orthogonality of vectors: two vectors are said to be orthogonal if their inner product is zero \[\mathbf{x} \cdot \mathbf{y} =\displaystyle\sum_{i=1}^{n}x_i\cdot y_i = 0\]

6.3 Null and unit vector

  • a null vector is a vector whose elements are all \(0\); the difference between any vector and itself yields a null vector
  • a unit vector is a vector whose elements are all \(1\)

Exercises

Exercise 6.1 Based on vector definitions and operations:

  1. find the vector \(\mathbf{x} + \mathbf{y}\) when \(\mathbf{x} =\begin{bmatrix} 1 \\ 2 \\ 5 \end{bmatrix}\) and \(\mathbf{y} =\begin{bmatrix} 0\\ 3 \\ 1 \end{bmatrix}\)

  2. find the vector \(2\mathbf{x} - \mathbf{y}\) when \(\mathbf{x} =\begin{bmatrix} -2 \\ 3 \\ 5 \end{bmatrix}\) and \(\mathbf{y} =\begin{bmatrix} 0\\ -4 \\ 7 \end{bmatrix}\)

  3. are \(\mathbf{u}\) and \(\mathbf{v}\) vectors orthogonal when when \(\mathbf{u} =\begin{bmatrix} 1 \\ 2 \end{bmatrix}\) and \(\mathbf{v} =\begin{bmatrix} 2\\ -1 \end{bmatrix}\)?

  4. are \(\mathbf{u}\) and \(\mathbf{v}\) vectors orthogonal when when \(\mathbf{u} =\begin{bmatrix} 3 \\ -1 \end{bmatrix}\) and \(\mathbf{v} =\begin{bmatrix} 7\\ 5 \end{bmatrix}\)?

  5. find the value \(n\) such that the vectors \(\mathbf{u} =\begin{bmatrix} 2 \\ 4 \\ 1 \end{bmatrix}\) and \(\mathbf{v} =\begin{bmatrix} n\\ 1 \\ 8 \end{bmatrix}\) are orthogonal.

Answers

Solution. Exercise 6.1

\[\mathbf{x} + \mathbf{y} = \begin{bmatrix} 1 \\ 2 \\ 5 \end{bmatrix} + \begin{bmatrix} 0 \\ 3 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 + 0\\ 2 + 3 \\ 5 + 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 5 \\ 6 \end{bmatrix}\]

  1. \[2\mathbf{x} - \mathbf{y} = \begin{bmatrix} 2 \cdot (-2) \\ 2 \cdot 3 \\ 2 \cdot 5 \end{bmatrix} + \begin{bmatrix} (-1) \cdot 0 \\ (-1) \cdot (-4) \\ (-1) \cdot 7 \end{bmatrix} = \begin{bmatrix} -4 \\ 6 \\ 10 \end{bmatrix} + \begin{bmatrix} 0 \\ 4 \\ -7 \end{bmatrix} = \begin{bmatrix} -4 + 0 \\ 6 + 4 \\ 10 - 7 \end{bmatrix} = \begin{bmatrix} -4 \\ 10 \\ 3 \end{bmatrix}\]

  2. Yes, to check orthogonality we need to calculate the inner product of two vectors and see if it is equal to 0, here \(\mathbf{u} \cdot \mathbf{v} =\displaystyle\sum_{i=1}^{2}u_i\cdot v_i = 1 \cdot 2 + 2 \cdot (-1) = 2 - 2 = 0\)

  3. No, since the inner product does not equal to 0 \[\mathbf{u} \cdot \mathbf{v} =\displaystyle\sum_{i=1}^{2}u_i\cdot v_i = 3 \cdot 7 + (-1) \cdot 5 = 21 - 5 = 16 \neq 0\]