Page 611 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

But it follows from 3 that of each side of the result and using 2a

or, equivalently,

Now, taking of each side of this equation and then
yields (verify)

or, equivalently,

In words, part (b) of Theorem 8.3.3 states that the inverse of a composition is the composition of the inverses in the reverse
order. This result can be extended to compositions of three or more linear transformations; for example,

(4)

In the case where , , and are matrix operators on , Formula 4 can be written as

which we might also write as

(5)

In words, this formula states that the standard matrix for the inverse of a composition is the product of the inverses of the
standard matrices of the individual operators in the reverse order.

Some problems that use Formula 5 are given in the exercises.

Dimension of Domain and Codomain

In Exercise 16 you are asked to show the important fact that if V and W are finite-dimensional vector spaces with

, and if is a linear transformation, then T cannot be one-to-one. In other words, the dimension

of the codomain W must be at least as large as the dimension of the domain V for there to be a one-to-one linear
transformation from V to W. This means, for example, that there can be no one-to-one linear transformation from space to

the plane .

EXAMPLE 8 Dimension and One-to-One Linear Transformations
A linear transformation from the plane to the real line R has a standard matrix

If is a point in , its image is

which is a scalar. But if , say, then there are infinitely many other points in that also have , since

there are infinitely many points on the line

This is because if a and b are nonzero, then every point of the form