38 CHAPTER 2. LINEAR TRANSFORMATIONS

because they are different sizes. As noted above, you write (cij) for the matrix C whoseijth entry is cij . In doing arithmetic with matrices you must define what happens in termsof the cij sometimes called the entries of the matrix or the components of the matrix.

The above discussion stated for general matrices is given in the following definition.

Definition 2.1.1 Let A = (aij) and B = (bij) be two m × n matrices. Then A + B = Cwhere

C = (cij)

for cij = aij + bij . Also if x is a scalar,

xA = (cij)

where cij = xaij . The number Aij will typically refer to the ijth entry of the matrix A. Thezero matrix, denoted by 0 will be the matrix consisting of all zeros.

Do not be upset by the use of the subscripts, ij. The expression cij = aij + bij is justsaying that you add corresponding entries to get the result of summing two matrices asdiscussed above.

Note that there are 2 × 3 zero matrices, 3 × 4 zero matrices, etc. In fact for every sizethere is a zero matrix.

With this definition, the following properties are all obvious but you should verify all ofthese properties are valid for A, B, and C, m× n matrices and 0 an m× n zero matrix,

A+B = B +A, (2.1)

the commutative law of addition,

(A+B) + C = A+ (B + C) , (2.2)

the associative law for addition,A+ 0 = A, (2.3)

the existence of an additive identity,

A+ (−A) = 0, (2.4)

the existence of an additive inverse. Also, for α, β scalars, the following also hold.

α (A+B) = αA+ αB, (2.5)

(α+ β)A = αA+ βA, (2.6)

α (βA) = αβ (A) , (2.7)

1A = A. (2.8)

The above properties, 2.1 - 2.8 are known as the vector space axioms and the fact thatthe m×n matrices satisfy these axioms is what is meant by saying this set of matrices withaddition and scalar multiplication as defined above forms a vector space.

Definition 2.1.2 Matrices which are n × 1 or 1 × n are especially called vectors and areoften denoted by a bold letter. Thus

x =

x1...

xn

is an n × 1 matrix also called a column vector while a 1 × n matrix of the form (x1 · · ·xn)is referred to as a row vector.