Consider the following equivalent columns.
Our answer to 1? is the matrix multiplication. What should be the answer to 2? and 3??
From the explicit formula for Ax, it is easy to see that
(A + B)x = Ax + Bx, (cA)x = c(Ax).
Therefore the answer to 2? is the following definition,
For linear transformations T, S: Rn → Rm, their addition T + S is defined by the formula (T + S)(x) = T(x) + S(x)
and the answer to 3? is the following definition.
For a linear transformation T: Rn → Rm, and a number r, the scalar multiplication rT is defined by the formula (rT)(x) = r(T(x))
Example Let T and S be the reflection in x-axis and the rotation by 90 degrees on the plane R2. The following picture illustrates the addition T + S.
The formula for the addition is (T + S)(x, y) = T(x, y) + S(x, y) = (x, - y) + (- y, x) = (x - y, x - y), and the corresponding matrix is
|[||1||0||] + [||0||-1||] = [||1||-1||]|
For another example, consider two transformations T = Projection to x-axis and S = Projection to y-axis from R2 to itself. The following picture shows their addition is the identity.
In terms of the corresponding matrix, we have
|[||1||0||] + [||0||0||] = [||1||0||]|
When the multiplication is mixed with addition and scalar multiplication, we have the following properties.
The properties are easy to prove in terms of linear transformations. For example, the following argument proves (T + S)R = TR + SR.
((T + S)R)(x)
= (T + S)(R(x)) (definition of composition)
= T(R(x)) + S(R(x)) (definition of addition)
= (TR)(x) + (SR)(x) (definition of composition)
= (TR + SR)(x).
The other properties can be proved in similar way.