Introduction to Matrices (Pt. 2)

Types of entities

Example	Math/Physics	Computer Science
[1]	Scalar (1-D)	Number (0-D)
[1, 2]	Vector (2-D)	Array (1-D)
[1, 2] [3, 4]	Matrix (3-D)	Matrix (2-D)
[[1, 2], [3,4]]	Tensor (N-D)	Tensor (N-D)

Dimensions

Poh-tay-toh, poh-tah-toh 🥔... ...

Toh-may-toh, toh-mah-toh 🍅... ...

With all confusing jargons aside, dimensions can be a difficult concept to wrap around. Despite having 7D dried mangoes and even 9D cinemas, it is difficult (actually impossible 🤪) to visualize even 4 dimensions.

You might have also noticed the difference in the amount of dimensions referred to by either via the Math/Physics vs Computer Science view.

This is because Math/Physics often work with what they refer to as the Vector Space (don't really claim to know this very well). Thus, the dimensions being referred to are the tools for the plotting of 2-D to 3-D vectors, to better understand topics such as forces of either push, pull, shear or rotation.

However, for Computer Science, the dimensions referred to here are often just a description about the arrays. Hence, instead of the former key purpose of plotting and visualization, this simply describes the level of complexity of a given array.

E.g., it may be helpful to describe the position of an element of a 2-D array (matrix) via both its row and column number, whereas it is not that necessary for a scalar (unnecessarily wrong)

Following this GitBook will be focusing on matrices, but however, the following entities are also a good mentioning/starting points.

Scalar

Scalar

A scalar is a single-value number. This entity is often applied to a vector, and its action is called scaling.

• For example, the value 3 is a scalar

• It should also not be represented with brackets

• It has 0 dimensions, as it is not meaningful to express it in terms of either rows or columns

• There are also no other possible variations to express a scalar

Vector

Vector

In Physics, a vector has both the characteristic of a magnitude and a direction. A vector of [2, 7] is able to tell the direction from the way it points, from the origin [0, 0] to [2, 7], and the magnitude through computing its Euclidean Distance (or alternatively the hypotenuse using Pythagoras Theorem).

In contrast to a scalar, a vector consist of

• More than one element, and

• There are two possible variation of expression

Row Vector

$$\mathbf{v_r } = \left[\begin{array} {rrr} x_1 & x_2 & x_3 \end{array}\right]$$

A row vector has the size of 1 x n.

Column Vector

$$\mathbf{v_c} = \left[\begin{array} {rrr} x_1\\ x_2\\ x_3 \end{array}\right]$$

A column vector has the size of m x 1.

More on the next page ⏭