Introduction to Linear Algebra

In this module, you will be introduced to some fundamental concepts in linear algebra that form the foundation of many data science and AI models you will encounter as you progress through your study program. Rather than teach it as a core mathematical topic, the aim of this module is to provide you with exposure to key topics in linear algebra that are essential for data science.

The study of linear algebra essentially involves the study of 3 mathematical objects - scalars, vectors, and matrices. Further, you will learn how to perform some elementary operations on these objects.

Today's learning objectives

  • Understand what is a scalar.
  • Understand what are vectors.
  • Understand what are matrices.
  • Implement the concepts learned in Python.

Scalars

A scalar, unlike vectors and matrices (which are arrays of numbers), is just a single number. The general notation is represent a scalar is to use lower-case variable names. For example, if you have a table of 10 items, you can represent the number of items using a scalar. Using mathematical notation, you would say something along the lines of "Let n ∈ N be the number of items in this table". And then assign the value 10 to n.

Vectors

A vector is simply just an array of numbers where the numbers are ordered by index. We can identify each individual number by its index in that ordering. Typically vectors are denoted using bold lower-cased letters, such as x. The elements of the vector are identified by writing its name in italic typeface, with a subscript. The first element of x is x1 , the second element is x2 and so on.


Please watch the following video to learn more about vectors and how to visualise vectors in your mind.

Matrix

A matrix is a 2-D array of numbers, so each element is identified by two indices (or dimensions) instead of just one. Vectors can be thought of as matrices that contain only one column. We usually give matrices upper-case variable names with bold typeface, such as A. If a real-valued matrix A has a height of m and a width of n, then we say that A ∈ Rm×n. We usually identify the elements of a matrix using its name in italic but not bold font, and the indices are listed with separating commas.

For example, in the matrix below, A1,1 is the upper left entry of A and A2,2 is the bottom right entry of A.


In general:

  1. A1,1 is always the upper left entry of A.
  2. Am,n is always the bottom right entry of A.
  3. Ai,i is always the diagonal entries of A where .

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