the entry in the \(\nth[i]\) row and \(\nth[j]\) column of a matrix
size (or dimensions) of a matrix
the number of rows and columns in a matrix, usually written \(m \times n\) to mean \(m\) rows and \(n\) columns
equal matrices
matrices with the same size, and the same numbers in corresponding entries
matrix addition
the process of creating a new matrix from two old matrices of identical sizes by adding corresponding entries
scalar multiple
the new matrix obtained from an old matrix obtained by multiplying every entry by the same number \(k\text{;}\) the common scale factor \(k\) is called a scalar
Remark4.2.1.
We will encounter the geometric origin of the word scalar in Chapter 11.
linear combination of matrices
a sum of scalar multiples of matrices; for example,
where \(k_1,k_2,\dotsc,k_m\) is a collection of scalars (numbers) and \(A_1,A_2,\dotsc,A_m\) is a collection of matrices all of the same dimensions
zero matrix
a matrix where every entry is zero, written \(\zerovec\)
row vector
a matrix consisting of a single row
column vector
a matrix consisting of a single column
vector of unknowns
a column vector containing all of the variables in a system
vector of constants
a column vector containing all of the constants from the right-hand sides of the equations in a system
vector form (for a system’s general solution)
a vector expression for the parametric form of the general solution to a system of equations in the form of a linear combination of column vectors, where the parameters are scalars in the linear combination
square matrix
a matrix with the same number of columns as rows
main diagonal
the diagonal of entries in a square matrix from top left to bottom right
transpose
the new matrix obtained from an old matrix by turning rows into columns and columns into rows; we usually write \(\utrans{A}\) to mean the transpose of the matrix \(A\)
