By 2.13, if A is an m × n matrix, then for v,u vectors in F^{n} and a,b scalars,
 (2.19) 
Definition 2.3.1 A function, A : F^{n} → F^{m} is called a linear transformation if for all u,v ∈ F^{n} and a,b scalars, 2.19 holds.
From 2.19, matrix multiplication defines a linear transformation as just defined. It turns out this is the only type of linear transformation available. Thus if A is a linear transformation from F^{n} to F^{m}, there is always a matrix which produces A. Before showing this, here is a simple definition.
Definition 2.3.2 A vector, e_{i} ∈ F^{n} is defined as follows:

where the 1 is in the i^{th} position and there are zeros everywhere else. Thus

Of course the e_{i} for a particular value of i in F^{n} would be different than the e_{i} for that same value of i in F^{m} for m≠n. One of them is longer than the other. However, which one is meant will be determined by the context in which they occur.
These vectors have a significant property.
Lemma 2.3.3 Let v ∈ F^{n}. Thus v is a list of numbers arranged vertically, v_{1},
 (2.20) 
Also, if A is an m × n matrix, then letting e_{i} ∈ F^{m} and e_{j} ∈ F^{n},
 (2.21) 
Proof: First note that e_{i}^{T} is a 1 × n matrix and v is an n × 1 matrix so the above multiplication in 2.20 makes perfect sense. It equals

as claimed.
Consider 2.21. From the definition of matrix multiplication, and noting that

by the first part of the lemma. ■
Theorem 2.3.4 Let L : F^{n} → F^{m} be a linear transformation. Then there exists a unique m×n matrix A such that

for all x ∈ F^{n}. The ik^{th} entry of this matrix is given by
 (2.22) 
Stated in another way, the k^{th} column of A equals Le_{k}.
Proof: By the lemma,

Let A_{ik} = e_{i}^{T}Le_{k}, to prove the existence part of the theorem.
To verify uniqueness, suppose Bx = Ax = Lx for all x ∈ F^{n}. Then in particular, this is true for x = e_{j} and then multiply on the left by e_{i}^{T} to obtain

showing A = B. ■
Corollary 2.3.5 A linear transformation, L : F^{n} → F^{m} is completely determined by the vectors
Proof: This follows immediately from the above theorem. The unique matrix determining the linear transformation which is given in 2.22 depends only on these vectors. ■
For a different proof of this theorem and corollary, see the following section.
This theorem shows that any linear transformation defined on F^{n} can always be considered as matrix multiplication. Therefore, the terms “linear transformation” and “matrix” are often used interchangeably. For example, to say that a matrix is one to one, means the linear transformation determined by the matrix is one to one.
Example 2.3.6 Find the linear transformation, L : ℝ^{2} → ℝ^{2} which has the property that Le_{1} =
