definition of canonical representative matrix of linear map between finite-product-of-copies-of-field vectors spaces
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of %field name% vectors space.
- The reader knows a definition of linear map.
- The reader knows a definition of %ring name% matrices space.
Target Context
- The reader will have a definition of canonical representative matrix of linear map between finite-product-of-copies-of-field vectors spaces.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( F\): \(\in \{\text{ the fields }\}\)
\( d_1\): \(\in \mathbb{N} \setminus \{0\}\)
\( d_2\): \(\in \mathbb{N} \setminus \{0\}\)
\( F^{d_1}\): \(= \text{ the } d_1 \text{ -dimensional } F \text{ vectors space }\)
\( F^{d_2}\): \(= \text{ the } d_2 \text{ -dimensional } F \text{ vectors space }\)
\( B_1\): \(= \{b_{1, 1}, ..., b_{1, d_1}\}\), \(= \text{ the canonical basis for } F^{d_1}\)
\( B_2\): \(= \{b_{2, 1}, ..., b_{2, d_2}\}\), \(= \text{ the canonical basis for } F^{d_2}\)
\( f\): \(: F^{d_1} \to F^{d_2}\), \(\in \{\text{ the linear maps }\}\)
\(*M\): \(= \begin{pmatrix} f (b_{1, 1})^1 & ... & f (b_{1, d_1})^1 \\ ... \\ f (b_{1, 1})^{d_2} & ... & f (b_{1, d_1})^{d_2} \end{pmatrix}\)
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Conditions:
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"the canonical basis" means that \(b_{l, j} = (0, ..., 0, 1, 0, ..., 0)^t\) where \(1\) is the \(j\)-th component.
2: Note
\(M\) indeed represents \(f\), by the proposition that for any map between any finite-product-of-copies-of-field vectors spaces, the map is linear if and only if the map is represented by the matrix with respect to the canonical bases.
