Linear Algebra, Theory And Applications, 2012a

2.3. LINEAR TRANSFORMATIONS 55
Proof: By the lemma,
(Lx) i
= e T i Lx = e T i x k Le k = ( e T i Le k
)
xk .
Let A ik = e T i 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 T i to obtain
showing A = B.
B ij = e T i Be j = e T i Ae j = A ij
Corollary 2.3.5 A linear transformation, L : F n → F m is completely determined by the
vectors {Le 1 , ··· ,Le n } .
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.
This theorem shows that any linear transformation defined on F n can always be considered
as a matrix. 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
2 1
Le 1 = and Le
1
2 = . From the above theorem and corollary, this linear transformation
is that determined by matrix multiplication by the matrix
3
( ) 2 1
.
1 3
Definition 2.3.7 Let L : F n → F m be a linear transformation and let its matrix be the
m × n matrix A. Then ker (L) ≡{x ∈ F n : Lx = 0} . Sometimes people also write this as
N (A) , the null space of A.
Then there is a fundamental result in the case where m