2.3. LINEAR TRANSFORMATIONS 55

where the 1 is in the ith position and there are zeros everywhere else. Thus

ei = (0, · · · , 0, 1, 0, · · · , 0)T .

Of course the ei for a particular value of i in Fn would be different than the ei for thatsame value of i in Fm for m ̸= n. One of them is longer than the other. However, which oneis meant will be determined by the context in which they occur.

These vectors have a significant property.

Lemma 2.3.3 Let v ∈ Fn. Thus v is a list of numbers arranged vertically, v1, · · · , vn. Then

eTi v = vi. (2.20)

Also, if A is an m× n matrix, then letting ei ∈ Fm and ej ∈ Fn,

eTi Aej = Aij (2.21)

Proof: First note that eTi is a 1 × n matrix and v is an n × 1 matrix so the abovemultiplication in 2.20 makes perfect sense. It equals

(0, · · · , 1, · · · 0)



v1...

vi...

vn

= vi

as claimed.Consider 2.21. From the definition of matrix multiplication, and noting that (ej)k = δkj

eTi Aej = eTi



∑k A1k (ej)k

...∑k Aik (ej)k

...∑k Amk (ej)k

= eTi



A1j

...

Aij

...

Amj

= Aij

by the first part of the lemma. ■

Theorem 2.3.4 Let L : Fn → Fm be a linear transformation. Then there exists a uniquem× n matrix A such that

Ax = Lx

for all x ∈ Fn. The ikth entry of this matrix is given by

eTi Lek (2.22)

Stated in another way, the kth column of A equals Lek.

Proof: By the lemma,

(Lx)i = eTi Lx = eTi∑k

xkLek =∑k

(eTi Lek

)xk.