Standard Matrix of a Linear Transformation from to
In Matrix Transformations and Introduction to Linear Transformations we learned several important properties of matrix transformations of and subspaces of . Let’s summarize the main points.
For a matrix transformation , induced by an matrix we have the following
results:
- is linear. (Theorem th:matrixtran) This means that for vectors and in and scalars and in .
- Columns of are the images of the standard unit vectors of under . (Observation obs:imagesOfijk)
- The action of on all of the elements of is completely determined by where maps the standard unit vectors. (See Examples ex:imageOfBasisVectors and ex:imageofatransformation)
The last point in the summary is so important that it is worth illustrating again.
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 Example ex:imageofatransformation, there was nothing special about the vector . Any vector of can be written as a unique linear combination of the standard unit vectors . Therefore, the image of any vector under a linear transformation is uniquely determined by the images of . Knowing allows us to construct a matrix , with as columns, that induces transformation . We formalize this idea in a theorem.
- Proof
- Observe that Because is linear, we have Thus, for every in , we have .
Theorem th:matrixtran shows that every matrix transformation is linear. Theorem th:matlin states that every linear transformation from into is a matrix transformation. We combine these results in a corollary.
The results of this section rely 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 Matrix of a Linear Transformation with Respect to Arbitrary Bases 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.
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 :Practice Problems
Problems prob:standardmatrix1-prob:standardmatrix4 Find the standard matrix of each linear transformation described below.