Data Science in Theory and Practice. Maria Cristina Mariani

      Definition 2.2 (Scaler multiplication of vectors) The product of a scalar

, and a vector is the vector obtained by multiplying each entry in the vector by the scalar:

Definition 2.3 (Vector addition) The sum of two vectors of the same size is the vector obtained by adding corresponding entries in the vectors:

      so that

is the vector with the

th element

.

      

      2.2.2 Matrices

      Definition 2.4 (Matrix) Let

and

denote positive integers. An

‐by‐

matrix is a rectangular array of real numbers with

rows and

columns:

      The notation

denotes the entry in row

, column

of

. In other words, the first index refers to the row number and the second index refers to the column number.

       Example 2.1

      then

.

      Definition 2.5 (Transpose of a matrix) The transpose operation

of a matrix changes the columns into rows, i.e. in matrix notation

, where “

” denotes transpose.

       Example 2.2

Definition 2.6 (Scaler multiplication of a matrix) The product of a scalar

, and a matrix is the matrix obtained by multiplying each entry in the matrix by the scalar:

      In other words,

.

      Definition 2.7 (Matrix addition) The sum of two vectors of the same size is the vector obtained by adding corresponding entries in the vectors:

      In other words,

.

      Definition 2.8 (Matrix multiplication) Suppose