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↑  "Linear transformations of  V into  V are often called linear operators on  V." Rudin 1976, tr. 207
↑  Let  V and  W be two real vector spaces. A mapping a from  V into  W Is called a 'linear mapping' or 'linear transformation' or 'linear operator' [...] from  V into  W, if





a
(
u
+
v
)
=
a
u
+
a
v


{\textstyle a(u+v)=au+av}

 for all 



u
,
v
∈
V


{\textstyle u,v\in V}

, 



a
(
λ
u
)
=
λ
a
u


{\textstyle a(\lambda u)=\lambda au}

 for all  u\in V and all real λ. Bronshtein & Semendyayev 2004, tr. 316Lỗi harv: nhiều mục tiêu (2×): CITEREFBronshteinSemendyayev2004 (trợ giúp)
↑ Rudin 1991, tr. 14 Lỗi harv: không có mục tiêu: CITEREFRudin1991 (trợ giúp) Here are some properties of linear mappings Λ Λ : X → → Y {\textstyle \Lambda :X\to Y} whose proofs are so easy that we omit them; it is assumed that A ⊂ ⊂ X {\textstyle A\subset X} and B ⊂ ⊂ Y {\textstyle B\subset Y} :
Λ
0
=
0.


{\textstyle \Lambda 0=0.}
If A is a subspace (or a convex set, or a balanced set) the same is true of 



Λ
(
A
)


{\textstyle \Lambda (A)}
If B is a subspace (or a convex set, or a balanced set) the same is true of 




Λ

−
1


(
B
)


{\textstyle \Lambda ^{-1}(B)}
In particular, the set:
 \Lambda ^{-1}(\{0\})=\{x\in X:\Lambda x=0\}={N}(\Lambda )

is a subspace of X, called the null space of 



Λ


{\textstyle \Lambda }

.
↑  Rudin 1991, tr. 14Lỗi harv: không có mục tiêu: CITEREFRudin1991 (trợ giúp). Suppose now that X and Y are vector spaces over the same scalar field. A mapping 



Λ
:
X
→
Y


{\textstyle \Lambda :X\to Y}

 is said to be linear if 



Λ
(
α
x
+
β
y
)
=
α
Λ
x
+
β
Λ
y


{\textstyle \Lambda (\alpha x+\beta y)=\alpha \Lambda x+\beta \Lambda y}

 for all 



x
,
y
∈
X


{\textstyle x,y\in X}

 and all scalars 



α


{\textstyle \alpha }

 and 



β


{\textstyle \beta }

. Note that one often writes 



Λ
x


{\textstyle \Lambda x}

, rather than 



Λ
(
x
)


{\textstyle \Lambda (x)}

, when 



Λ


{\textstyle \Lambda }

 is linear.
↑  Rudin 1976, tr. 206. A mapping A of a vector space X into a vector space Y is said to be a linear transformation if: 



A

(



x


1


+


x


2



)

=
A


x


1


+
A


x


2


,
 
A
(
c

x

)
=
c
A

x



{\textstyle A\left(\mathbf {x} _{1}+\mathbf {x} _{2}\right)=A\mathbf {x} _{1}+A\mathbf {x} _{2},\ A(c\mathbf {x} )=cA\mathbf {x} }

 for all 




x

,


x


1


,


x


2


∈
X


{\textstyle \mathbf {x} ,\mathbf {x} _{1},\mathbf {x} _{2}\in X}

 and all scalars c. Note that one often writes 



A

x



{\textstyle A\mathbf {x} }

 instead of 



A
(

x

)


{\textstyle A(\mathbf {x} )}

 if A is linear.
↑  Rudin 1991, tr. 14Lỗi harv: không có mục tiêu: CITEREFRudin1991 (trợ giúp). 
Linear mappings of X onto its scalar field are called linear functionals.
↑  https://math.stackexchange.com/a/62791/401895
↑  Wilansky 2013, tr. 21-26.Lỗi sfn: không có mục tiêu: CITEREFWilansky2013 (trợ giúp)
↑  Rudin 1976, tr. 210

Suppose 




{



x


1


,
…
,


x


n



}



{\textstyle \left\{\mathbf {x} _{1},\ldots ,\mathbf {x} _{n}\right\}}

 and 




{



y


1


,
…
,


y


m



}



{\textstyle \left\{\mathbf {y} _{1},\ldots ,\mathbf {y} _{m}\right\}}

 are bases of vector spaces X and Y, respectively. Then every 



A
∈
L
(
X
,
Y
)


{\textstyle A\in L(X,Y)}

 determines a set of numbers 




a

i
,
j




{\textstyle a_{i,j}}

 such that

 A\mathbf {x} _{j}=\sum _{i=1}^{m}a_{i,j}\mathbf {y} _{i}\quad (1\leq j\leq n).

It is convenient to represent these numbers in a rectangular array of  m rows and  n columns, called an  m by  n matrix:

 [A]={\begin{bmatrix}a_{1,1}&a_{1,2}&\ldots &a_{1,n}\\a_{2,1}&a_{2,2}&\ldots &a_{2,n}\\\vdots &\vdots &\ddots &\vdots \\a_{m,1}&a_{m,2}&\ldots &a_{m,n}\end{bmatrix}}

Observe that the coordinates 




a

i
,
j




{\textstyle a_{i,j}}

 of the vector 



A



x



j




{\textstyle A{\mathbf {x} }_{j}}

 (with respect to the basis 



{


y


1


,
…
,


y


m


}


{\textstyle \{\mathbf {y} _{1},\ldots ,\mathbf {y} _{m}\}}

) appear in the jth column of 



[
A
]


{\textstyle [A]}

. The vectors 



A



x



j




{\textstyle A{\mathbf {x} }_{j}}

 are therefore sometimes called the column vectors of 



[
A
]


{\textstyle [A]}

. With this terminology, the range of A is spanned by the column vectors of 



[
A
]


{\textstyle [A]}

.
↑  Axler, Sheldon (2015). Linear Algebra Done Right. Undergraduate Texts in Mathematics (ấn bản thứ 3). Springer Publishing. tr. 52. ISBN 978-3-319-11079-0. ISSN 0172-6056.
↑  Tu, Loring (2011). An Introduction to Manifolds. Universitext (ấn bản thứ 2). Springer. tr. 19. ISBN 978-1-4419-7399-3. ISSN 0172-5939.
↑  Horn & Johnson 2013, 0.2.3 Vector spaces associated with a matrix or linear transformation, p. 6
↑  Katznelson, Yitzhak; Katznelson, Yonatan R. (2008). A (Terse) Introduction to Linear Algebra. American Mathematical Society. tr. 52. ISBN 978-0-8218-4419-9.