Linear Algebra, Theory And Applications, 2012a

54 MATRICES AND LINEAR TRANSFORMATIONS
Definition 2.3.2 Avector,e i ∈ F n is defined as follows:
⎛ ⎞
0
.
e i ≡
1
,
⎜ ⎟
⎝
. ⎠
0
where the 1 is in the i th position and there are zeros everywhere else. Thus
e i =(0, ··· , 0, 1, 0, ··· , 0) T .
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 , ··· ,v n . Then
Also, if A is an m × n matrix, then letting e i ∈ F m and e j ∈ F n ,
e T i v = v i . (2.20)
e T i Ae j = A ij (2.21)
Proof: First note that e T i is a 1 × n matrix and v is an n × 1 matrix so the above
multiplication in (2.20) makes perfect sense. It equals
⎛ ⎞
v 1
.
(0, ··· , 1, ···0)
v i
= v i
⎜ ⎟
⎝
. ⎠
v n
as claimed.
Consider (2.21). From the definition of matrix multiplication, and noting that (e j ) k
=
δ kj
e T i Ae j = e T i
⎛
⎜
⎝
by the first part of the lemma.
∑
k A 1k (e j ) k
.
∑
k A ik (e j ) k
.
∑
k A mk (e j ) k
⎞ ⎛
= e T i
⎟ ⎜
⎠ ⎝
A 1j
.
A ij
.
A mj
⎞
= A ij
⎟
⎠
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
Ax = Lx
for all x ∈ F n . The ik th entry of this matrix is given by
Stated in another way, the k th column of A equals Le k .
e T i Le k (2.22)