Data representation

In this section, we will look at core data structures and representations used most commonly across different linear algebra tasks. This is not meant to be a comprehensive list at all but only serves to highlight some of the prominent representations useful for understanding deep learning concepts:

• Vectors: One of the most fundamental representations in linear algebra is a vector. A vector can be defined as an array of objects, or more specifically an array of numbers that preserves the ordering of the numbers. Each number can be accessed in a vector based on its indexed location. For example, consider a vector x containing seven days a week encoded from 1 to 7, where 1 represents Sunday and 7 represents Saturday. Using this notation, a particular day of the week, say Wednesday, can be directly accessed from the vector as x [4]:

• Matrices: These are a two-dimensional representation of numbers, or basically a vector of vectors. Each matrix, m, is composed of a certain number of rows, r, and a specified number of columns, c. Each of i rows, where 1 <= i <= r, is a vector of c numbers. Each of the j columns, where 1 <=j <= c, is also a vector of r numbers. Matrices are a particularly useful representation when we are working with images. Though real-world images are three-dimensional in nature, most of the computer vision problems are focused on the two-dimensional presentation of images. As such, a matrix representation is an intuitive representation of images:

• Identity matrices: An identity matrix is defined as a matrix which, when multiplied with a vector, does not change the vector. Typically, an identity matrix has all elements as 0 except on its main diagonal, which is all 1s: