Every $m\times n$ matrix defines a linear map $T_A:\mathbb F^n\to\mathbb F^m$ by $T_A(x)=Ax$. Conversely, every linear map between these coordinate spaces has one unique standard matrix. Columns encode images of basis vectors; matrix multiplication encodes composition.
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- Matrices act as a transformation by multiplying vectorsFor $A=[a 1\ \cdots\ a n]$, the action on $x$ is $Ax=\sum jx ja j$. Thus the map replaces each standard basis...
- Matrix-vector multiplication is a linear transformation.The map $T(x)=Ax$ is linear because $A(\alpha x+\beta y)=\alpha Ax+\beta Ay$ for all vectors and scalars. This...
- Definition of the standard matrix for a linear transformationThe standard matrix of $T:\mathbb F^n\to\mathbb F^m$ is $[T(e 1)\ \cdots\ T(e n)]$. It is the unique matrix $A$...
- A linear transformation is given by a matrix whose columns are the images of the standard basis vectorsSince $x=\sum jx je j$, linearity gives $T(x)=\sum jx jT(e j)$. Therefore $T(x)=[T(e 1)\ \cdots\ T(e n)]x$. This...
- A linear transformation is given by a matrix with respect to a given basis.For ordered bases $B=(b j)$ of $V$ and $C$ of $W$, define $[T] {C\leftarrow B}$ by taking column $j$ to be $[T(b j)]...
- Definition of nilpotent linear transformationA linear operator $T:V\to V$ is nilpotent if $T^k=0$ for some positive integer $k$. The least such $k$ is its...
- Definition of matrix representation of a linear transformationA matrix representation records a linear map using coordinates in chosen bases. If $T:V\to W$, $B$ is a basis of $V$,...
- Matrix representation of a composition of linear transformations is given by a matrix productFor $T:U\to V$ and $S:V\to W$, use compatible bases $B,C,D$. Then $[S\circ T] {D\leftarrow B}=[S] {D\leftarrow C}[T]...
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