Page 651 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

8.6 Our previous work shows that every real vector space of dimension n can be
related to through coordinate vectors and that every linear transformation
ISOMORPHISM from a real vector space of dimension n to one of dimension m can be related
to and through transition matrices. In this section we shall further
strengthen the connection between a real vector space of dimension n and .

Onto Transformations is onto if the range of T is W—that is, if for

Let V and W be real vector spaces. We say that the linear transformation
every w in W, there is a v in V such that

An onto transformation is also said to be surjective or to be a surjection. For a surjective mapping, then, the range and the
codomain coincide.

EXAMPLE 1 Onto Transformations

Consider the projection defined by . This is an onto mapping, because if is a point in

, then is mapped to it. (Of course, so are infinitely many other points in .)

Consider the transformation defined by . This is essentially the same as P except that we
. This mapping is not onto, because, for example, the point (1, 1,
consider the result to be a vector in rather than a vector in

1) in the codomain is not the image of any v in the domain.

If a transformation is both one-to-one (also called injective or an injection) and onto, then it is a one-to-one mapping

to its range W and so has an inverse . A transformation that is one-to-one and onto is also said to be bijective or to

be a bijection between V and W. In the exercises, you'll be asked to show that the inverse of a bijection is also a bijection.

In Section 8.3 it was stated that if V and W are finite-dimensional vector spaces, then the dimension of the codomain W must be at

least as large as the dimension of the domain V for there to exist a one-to-one linear transformation from V to W. That is, there

can be an injective linear transformation from V to W only if . Similarly, there can be a surjective linear

transformation from V to W only if . Theorem 8.6.1 follows immediately.

THEOREM 8.6.1

Bijective Linear Transformations , then there can be no bijective linear transformation

Let V and W be finite-dimensional vector spaces. If
from V to W.

Isomorphisms

Bijective linear transformations between vector spaces are sufficiently important that they have their own name.