Linear map (English Wikipedia)

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"Linear transformations of $V$ into $V$ are often called linear operators on $V$ ." Rudin 1976, p. 207 Rudin, Walter (1976). Principles of Mathematical Analysis. Walter Rudin Student Series in Advanced Mathematics (3rd ed.). New York: McGraw–Hill. ISBN 978-0-07-054235-8.
Rudin 1991, p. 14
Here are some properties of linear mappings ${\textstyle \Lambda :X\to Y}$ ${\textstyle \Lambda :X\to Y}$ whose proofs are so easy that we omit them; it is assumed that ${\textstyle A\subset X}$ ${\textstyle A\subset X}$ and ${\textstyle B\subset Y}$ ${\textstyle B\subset Y}$ :
1. ${\textstyle \Lambda 0=0.}$
2. Rudin 1991, p. 14. Suppose now that $X$ and $Y$ are vector spaces over the same scalar field. A mapping ${\textstyle \Lambda :X\to Y}$ is said to be linear if ${\textstyle \Lambda (\alpha \mathbf {x} +\beta \mathbf {y} )=\alpha \Lambda \mathbf {x} +\beta \Lambda \mathbf {y} }$ for all ${\textstyle \mathbf {x} ,\mathbf {y} \in X}$ and all scalars ${\textstyle \alpha }$ and ${\textstyle \beta }$ . Note that one often writes ${\textstyle \Lambda \mathbf {x} }$ , rather than ${\textstyle \Lambda (\mathbf {x} )}$ , when ${\textstyle \Lambda }$ is linear. Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
3. Rudin 1976, p. 206. A mapping $A$ of a vector space $X$ into a vector space $Y$ is said to be a linear transformation if: ${\textstyle A\left(\mathbf {x} _{1}+\mathbf {x} _{2}\right)=A\mathbf {x} _{1}+A\mathbf {x} _{2},\ A(c\mathbf {x} )=cA\mathbf {x} }$ for all ${\textstyle \mathbf {x} ,\mathbf {x} _{1},\mathbf {x} _{2}\in X}$ and all scalars $c$ . Note that one often writes ${\textstyle A\mathbf {x} }$ instead of ${\textstyle A(\mathbf {x} )}$ if $A$ is linear. Rudin, Walter (1976). Principles of Mathematical Analysis. Walter Rudin Student Series in Advanced Mathematics (3rd ed.). New York: McGraw–Hill. ISBN 978-0-07-054235-8.
4. Rudin 1991, p. 14. Linear mappings of $X$ onto its scalar field are called linear functionals. Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
5. Rudin 1976, p. 210 Suppose ${\textstyle \left\{\mathbf {x} _{1},\ldots ,\mathbf {x} _{n}\right\}}$ and ${\textstyle \left\{\mathbf {y} _{1},\ldots ,\mathbf {y} _{m}\right\}}$ are bases of vector spaces $X$ and $Y$ , respectively. Then every ${\textstyle A\in L(X,Y)}$ determines a set of numbers ${\textstyle a_{i,j}}$ 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 ${\textstyle a_{i,j}}$ of the vector ${\textstyle A\mathbf {x} _{j}}$ (with respect to the basis ${\textstyle \{\mathbf {y} _{1},\ldots ,\mathbf {y} _{m}\}}$ ) appear in the j^th column of ${\textstyle [A]}$ . The vectors ${\textstyle A\mathbf {x} _{j}}$ are therefore sometimes called the column vectors of ${\textstyle [A]}$ . With this terminology, the range of $A$ is spanned by the column vectors of ${\textstyle [A]}$ . Rudin, Walter (1976). Principles of Mathematical Analysis. Walter Rudin Student Series in Advanced Mathematics (3rd ed.). New York: McGraw–Hill. ISBN 978-0-07-054235-8.
6. Rudin 1991, p. 15 1.18 Theorem Let ${\textstyle \Lambda }$ be a linear functional on a topological vector space $X$ . Assume ${\textstyle \Lambda \mathbf {x} \neq 0}$ for some ${\textstyle \mathbf {x} \in X}$ . Then each of the following four properties implies the other three:
  1. ${\textstyle \Lambda }$

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Nistor, Victor (2001) [1994], "Index theory", Encyclopedia of Mathematics, EMS Press: "The main question in index theory is to provide index formulas for classes of Fredholm operators ... Index theory has become a subject on its own only after M. F. Atiyah and I. Singer published their index theorems"

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"terminology - What does 'linear' mean in Linear Algebra?". Mathematics Stack Exchange. Retrieved 2021-02-17.

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Rudin 1991, p. 14
Here are some properties of linear mappings ${\textstyle \Lambda :X\to Y}$ ${\textstyle \Lambda :X\to Y}$ whose proofs are so easy that we omit them; it is assumed that ${\textstyle A\subset X}$ ${\textstyle A\subset X}$ and ${\textstyle B\subset Y}$ ${\textstyle B\subset Y}$ :
1. ${\textstyle \Lambda 0=0.}$
2. Rudin 1991, p. 14. Suppose now that $X$ and $Y$ are vector spaces over the same scalar field. A mapping ${\textstyle \Lambda :X\to Y}$ is said to be linear if ${\textstyle \Lambda (\alpha \mathbf {x} +\beta \mathbf {y} )=\alpha \Lambda \mathbf {x} +\beta \Lambda \mathbf {y} }$ for all ${\textstyle \mathbf {x} ,\mathbf {y} \in X}$ and all scalars ${\textstyle \alpha }$ and ${\textstyle \beta }$ . Note that one often writes ${\textstyle \Lambda \mathbf {x} }$ , rather than ${\textstyle \Lambda (\mathbf {x} )}$ , when ${\textstyle \Lambda }$ is linear. Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
3. Rudin 1991, p. 14. Linear mappings of $X$ onto its scalar field are called linear functionals. Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
4. Wilansky 2013, pp. 21–26. Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.
5. Kubrusly 2001, p. 57. Kubrusly, Carlos (2001). Elements of operator theory. Boston: Birkhäuser. ISBN 978-1-4757-3328-0. OCLC 754555941.
6. Schechter 1996, pp. 277–280. Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.
7. Rudin 1991, p. 15 1.18 Theorem Let ${\textstyle \Lambda }$ be a linear functional on a topological vector space $X$ . Assume ${\textstyle \Lambda \mathbf {x} \neq 0}$ for some ${\textstyle \mathbf {x} \in X}$ . Then each of the following four properties implies the other three:
  1. ${\textstyle \Lambda }$