You are about to erase your work on this activity. Are you sure you want to do this?
Updated Version Available
There is an updated version of this activity. If you update to the most recent version of this activity, then your current progress on this activity will be erased. Regardless, your record of completion will remain. How would you like to proceed?
We establish that a linear transformation of a vector space is completely determined
by its action on a basis.
Note to Student: In this module we will often use , and to denote the domain and
codomain of linear transformations. If you are familiar with abstract vector spaces,
you can regard , and as finite-dimensional vector spaces, otherwise you may think of
, and as subspaces of .
LTR-0025: Linear Transformations and Bases
Recall that a transformation is called a linear transformation if the following are
true for all vectors and in , and scalars .
Suppose we want to define a linear transformation by
Is this information sufficient to define ? To answer this question we will try to
determine what does to an arbitrary vector of .
If is a vector in , then can be uniquely expressed as a linear combination of and
By linearity of we have
This shows that the image of any vector of under is completely determined by the
action of on the standard unit vectors and .
Vectors and form a standard basis of . What if we want to use a different
Let be our basis of choice for . (How would you verify that is a basis of ?) And
suppose we want to define a linear transformation by
Is this enough information to define ?
Because form a basis of , every element of can be written as a unique linear
We can find as follows:
Again, we see how a linear transformation is completely determined by its action on a
Our discussion thus far hinged on the fact that the representation of a vector in
terms of the given basis elements is unique. Imagine what would happen if this were
not the case. In Exploration init:tij, for instance, we might have been able to represent as
and ( or ). This would have potentially resulted in mapping to two different
elements: and , implying that is not even a function. Fortunately, Theorem th:linindbasis of
VSP-0030 assures us that in , vector representation in terms of the elements of a
basis is unique. Theorem th:uniquerep of VSP-0060 generalizes this result to abstract vector
Suppose we want to define a linear transformation . Let be a basis of . To define , it
is sufficient to state the image of each basis vector under . Once the images of the
basis vectors are established, we can determine the images of all vectors of as
Given any vector of , we can uniquely express as a linear combination of .
for some scalar coefficients . We find as follows:
In our final example, we will consider in the context of a basis of the codomain, as
well as a basis of the domain. This will later help us tackle the question
of the matrix of associated with bases other than the standard basis of
Because each of and is linearly independent, let
be bases of and , respectively.
Define a linear transformation by
Verify that is in .
item:subtosub1a We need to express as a linear combination of and . This can be done by
observation or by solving the equation
We find that and , so . Thus is in .
The important observation here is that given a linear transformation defined on the
basis elements of in terms of the basis elements of , we are able to find the image of
any in in terms of the basis elements of . We will visit this idea again in Exploration