18.1. THE MATRIX OF A LINEAR TRANSFORMATION 377

You can add matrices of the same size by adding the corresponding entries. Indeed, youmust do this if you want to preserve the idea that matrix multiplication of a vector gives alinear transformation of the vector. Say T,S ∈ L (Rn,Rm) . These are just functions of aspecial sort. Thus T +S is defined as the function which does the following: (T +S)(x)≡T (x)+S (x) .

∑k[T +S]ik xk ≡ ((T +S)(x))i ≡ (T (x)+S (x))i

= (T (x))i +(S (x))i = ([T ]x)i +([S]x)i

= ∑k[T ]ik xk +∑

k[S]ik xk = ∑

k([T ]ik +[S]ik)xk

Since x is arbitrary, it follows that [T +S]ik = [T ]ik +[S]ik . In other words, you must addcorresponding entries. This shows why you must add matrices of the same size. Similarlyyou need α [T ] = [αT ].

Then in terms scalar multiplication and addition of either matrices or linear transforma-tions, following properties are called the vector space axioms.

• Commutative Law Of Addition.

A+B = B+A, (18.2)

• Associative Law for Addition.

(A+B)+C = A+(B+C) , (18.3)

• Existence of an Additive Identity

A+0 = A, (18.4)

• Existence of an Additive Inverse

A+(−A) = 0, (18.5)

Also for α,β scalars, the following additional properties hold.

• Distributive law over Matrix Addition.

α (A+B) = αA+αB, (18.6)

• Distributive law over Scalar Addition

(α +β )A = αA+βA, (18.7)

• Associative law for Scalar Multiplication

α (βA) = αβ (A) , (18.8)

• Rule for Multiplication by 1.1A = A. (18.9)