↑ "Linear transformations of V {\displaystyle V} into V {\displaystyle V} are often called linear operators on V {\displaystyle V} ." Rudin 1976, tr. 207
↑ Let V {\displaystyle V} and W {\displaystyle W} be two real vector spaces. A mapping a from V {\displaystyle V} into W {\displaystyle W} Is called a 'linear mapping' or 'linear transformation' or 'linear operator' [...] from V {\displaystyle V} into W {\displaystyle 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 ∈ V {\displaystyle u\in V} and all real λ. Bronshtein & Semendyayev 2004, tr. 316Lỗi harv: nhiều mục tiêu (2×): CITEREFBronshteinSemendyayev2004 (trợ giúp)
↑ Rudin 1991, tr. 14Lỗi harv: không có mục tiêu: CITEREFRudin1991 (trợ giúp)Here are some properties of linear mappings 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: Λ − 1 ( { 0 } ) = { x ∈ X : Λ x = 0 } = N ( Λ ) {\displaystyle \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({\bf {{x}_{1}+{\bf {{x}_{2}}}}}\right)=A{\bf {{x}_{1}+A{\bf {{x}_{2},\ A(c{\bf {{x})=cA{\bf {x}}}}}}}}} for all x , x 1 , x 2 ∈ X {\textstyle {\bf {{x},{\bf {{x}_{1},{\bf {{x}_{2}\in X}}}}}}} and all scalars c. Note that one often writes A x {\textstyle A{\bf {x}}} instead of A ( x ) {\textstyle A({\bf {{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.
↑ Rudin 1976, tr. 210Suppose { x 1 , … , x n } {\textstyle \left\{{\bf {{x}_{1},\ldots ,{\bf {{x}_{n}}}}}\right\}} and { y 1 , … , y m } {\textstyle \left\{{\bf {{y}_{1},\ldots ,{\bf {{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 x j = ∑ i = 1 m a i , j y i ( 1 ≤ j ≤ n ) . {\displaystyle A{\bf {{x}_{j}=\sum _{i=1}^{m}a_{i,j}{\bf {{y}_{i}\quad (1\leq j\leq n).}}}}} It is convenient to represent these numbers in a rectangular array of m {\displaystyle m} rows and n {\displaystyle n} columns, called an m {\displaystyle m} by n {\displaystyle n} matrix: [ A ] = [ a 1 , 1 a 1 , 2 … a 1 , n a 2 , 1 a 2 , 2 … a 2 , n ⋮ ⋮ ⋱ ⋮ a m , 1 a m , 2 … a m , n ] {\displaystyle [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{\bf {x}}_{j}} (with respect to the basis { y 1 , … , y m } {\textstyle \{{\bf {{y}_{1},\ldots ,{\bf {{y}_{m}\}}}}}} ) appear in the jth column of [ A ] {\textstyle [A]} . The vectors A x j {\textstyle A{\bf {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]} .
