Matrices and Vectors

 

Matrices are 2-dimensional arrays:

[abcdefghijkl]\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ j & k & l\end{bmatrix}

The above matrix has four rows and three columns, so it is a 4 x 3 matrix.

A vector is a matrix with one column and many rows:

[wxyz]\begin{bmatrix} w \\ x \\ y \\ z \end{bmatrix}

So vectors are a subset of matrices. The above vector is a 4 x 1 matrix.

Notation and terms:

  • AijA_{ij} refers to the element in the iith row and jjth column of matrix AA.
  • AA vector with 'n' rows is referred to as an 'n'-dimensional vector.
  • viv_i refers to the element in the iith row of the vector.
  • In general, all our vectors and matrices will be 1-indexed. Note that for some programming languages, the arrays are 0-indexed.
  • Matrices are usually denoted by uppercase names while vectors are lowercase.
  • "Scalar" means that an object is a single value, not a vector or matrix.
  • R\mathbb{R} refers to the set of scalar real numbers.
  • Rn\mathbb{R^n} refers to the set of n-dimensional vectors of real numbers.
  • Rm×n\mathbb{R^{m \times n}} refers to the set of m×nm \times n matrices of real numbers.