alt="bold x equals Start 4 By 1 Matrix 1st Row x 1 2nd Row x 2 3rd Row vertical-ellipsis 4th Row x Subscript n Baseline EndMatrix period"/>
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
