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.
LTR-0020: Standard Matrix of a Linear Transformation from to
Columns of a Matrix as Images of Standard Unit Vectors
Let be a linear transformation induced by
We will examine the effect of on the standard unit vectors , and .
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 .
Linear Transformations of as Matrix Transformations
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.