Observe that the image of is the first column of , the image of is the second column of , and the image of is the third column.

We establish that every linear transformation of is a matrix transformation, and define the standard matrix of a linear transformation.

Let be a linear transformation induced by
We will examine the effect of on the standard unit vectors , and .

Observe that the image of is the first column of , the image of is the second column of , and the image of is the third column.

In general, the linear transformation , induced by an matrix maps the standard unit vectors to the columns of . We summarize this observation by expressing columns of as images of vectors under .

Recall that matrix transformations are linear (Theorem th:matrixtran of LTR-0010). We
now know that standard unit vectors map to the columns of the matrix
that induces the linear transformation. What we do not yet know is that
all linear transformations are induced by matrices. To show that all linear
transformations from into are matrix transformations, we will demonstrate that the
destination of the standard unit vectors under a linear transformation determines
the images of *all* vectors of under . We first illustrate this idea with an
example.

Let be a linear transformation. Suppose that the only information we have about
this transformation is that , and . Is this information sufficient to determine the
image of ?

In general, every vector of can be written as a unique linear combination of the standard unit vectors . Therefore, the image of every vector under a linear transformation is uniquely determined by the images of . Knowing allows us to construct a matrix that induces the desired linear transformation. We formalize this idea in a theorem.

Let be a linear transformation. Then
satisfies . In other words, induces the linear transformation .

- Proof
- Observe that
Because is linear, we have

Thus, for every in , we have .

The above discussion relies on the fact that every vector of can be written as a
unique linear combination of the standard unit vectors . These vectors form
the *standard basis* for . We will see in LTR-0080 that the matrix used to
represent a linear transformation depends on a choice of basis. Because we
are using the standard basis, it is natural to name the matrix in Theorem th:matlin
accordingly.

In Theorem th:matrixtran of LTR-0010 we showed that every transformation induced by a matrix is linear. Theorem th:matlin states that every linear transformation from into is induced by a matrix. We combine these results in a corollary.

Find the standard matrix of a linear transformation if and .

In this example we are
not given the images of the standard basis vectors and . However, we can find the
images of and by expressing and as linear combinations of and , then apply the
fact that is linear.

Let’s start with the easy one. Therefore, by linearity of , we have: This gives us the first column of the standard matrix for .

You can solve the vector equation to express as a linear combination of and as follows: By linearity of ,

This gives us the second column of the standard matrix. Putting all of the information together, we get the following standard matrix for :

Consider the matrix
In this problem we will investigate the geometric nature of the transformation
induced by the matrix . First, observe that . (Why?) We know that maps maps to ,
and maps to itself. But what does do to an arbitrary vector in ? To find out, we will
apply to an arbitrary vector of .
This computation shows that negates the component of a vector and leaves the
component unchanged. The diagram below shows the action of on several
vectors.

From a geometric perspective, negating the component while leaving the component unchanged, reflects all vectors in the plane about the -axis.