5.1 Linear Transformations and Matrices
We can also refer to a linear transformation as a linear function. These are defined as follows.
Definition 5.1.1 Let T be a function defined on Fn which takes vectors in Fn to vectors in Fm. This is
written as T : Fn → Fm. It is a linear function or equivalently linear transformation if it satisfies the
following: For a,b scalars and x,y vectors in Fn it follows that
In words: It goes across addition and you can factor out scalars. Then notice that an m×n matrix A has
the property that if x is in Fn then Ax is in Fm and by Theorem 5.0.5, if Tx ≡ Ax for A an m×n matrix,
then it follows that T is a linear function.
The following definition defines a linear function and notes that matrix multiplication gives an example
of such a thing. The next theorem shows that this is the only way it can happen.
Theorem 5.1.2 Let T be a linear transformation, T : Fn → Fm. Then there exists an m × n matrix A
such that for all x ∈ Fn, you have Tx = Ax. This matrix is given by
Proof: Let x be arbitrary and x =
It follows that, since T is linear,
and so the matrix which does what is claimed is the one whose ith column is Tei. That is