Biến đổi tuyến tính (Vietnamese Wikipedia)

Analysis of information sources in references of the Wikipedia article "Biến đổi tuyến tính" in Vietnamese language version.

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"Linear transformations of $V$ into $V$ are often called linear operators on $V$ ." Rudin 1976, tr. 207 Rudin, Walter (1976). Principles of Mathematical Analysis. Walter Rudin Student Series in Advanced Mathematics (ấn bản thứ 3). New York: McGraw–Hill. ISBN 978-0-07-054235-8.
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: ${\textstyle A\left(\mathbf {x} _{1}+\mathbf {x} _{2}\right)=A\mathbf {x} _{1}+A\mathbf {x} _{2},\ A(c\mathbf {x} )=cA\mathbf {x} }$ 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 (ấn bản thứ 3). New York: McGraw–Hill. ISBN 978-0-07-054235-8.
Rudin 1976, tr. 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 (ấn bản thứ 3). New York: McGraw–Hill. ISBN 978-0-07-054235-8.
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.
Katznelson, Yitzhak; Katznelson, Yonatan R. (2008). A (Terse) Introduction to Linear Algebra. American Mathematical Society. tr. 52. ISBN 978-0-8218-4419-9.