definition of representative matrix of linear map between finite-dimensional vectors spaces with respect to bases
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 basis of module.
- The reader knows a definition of %ring name% matrices space.
- The reader admits the proposition that for any finite-dimensional vectors space and any basis, the vectors space is 'vectors spaces - linear morphisms' isomorphic to the components vectors space with respect to the basis.
Target Context
- The reader will have a definition of representative matrix of linear map between finite-dimensional vectors spaces with respect to bases.
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:
\( F\): \(\in \{\text{ the fields }\}\)
\( d_1\): \(\in \mathbb{N} \setminus \{0\}\)
\( d_2\): \(\in \mathbb{N} \setminus \{0\}\)
\( V_1\): \(\in \{\text{ the } d_1 \text{ -dimensional } F \text{ vectors spaces }\}\)
\( V_2\): \(\in \{\text{ the } d_2 \text{ -dimensional } F \text{ vectors spaces }\}\)
\( B_1\): \(\in \{\text{ the bases for } V_1\}\)
\( B_2\): \(\in \{\text{ the bases for } V_2\}\)
\( f\): \(: V_1 \to V_2\), \(\in \{\text{ the linear maps }\}\)
\( f_1\): \(: V_1 \to F^{d_1}\), \(= \text{ the canonical 'vectors spaces - linear morphisms' isomorphism to the components vectors space with respect to } B_1\)
\( f_2\): \(: V_2 \to F^{d_2}\), \(= \text{ the canonical 'vectors spaces - linear morphisms' isomorphism to the components vectors space with respect to } B_2\)
\( f_2 \circ f \circ {f_1}^{-1}\): \(: F^{d_1} \to F^{d_2}\)
\(*M\): \(= \text{ the canonical representative matrix of } f_2 \circ f \circ {f_1}^{-1}\)
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Conditions:
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2: Note
\(f_1\) and \(f_2\) are by the proposition that for any finite-dimensional vectors space and any basis, the vectors space is 'vectors spaces - linear morphisms' isomorphic to the components vectors space with respect to the basis.
\(f_2 \circ f \circ {f_1}^{-1}\) is linear, by the proposition that for any linear map between any modules and any linear map from any supermodule of the codomain of the 1st map into any module, the composition of the 2nd map after the 1st map is linear.
So, \(M\) is well-defined by the definition of canonical representative matrix of linear map between finite-product-of-copies-of-field vectors spaces.
\(M\) is called "representative" because \(f\) can be reconstructed from \(M\): \(M\) determines \(f_2 \circ f \circ {f_1}^{-1}\) and \(f = {f_2}^{-1} \circ f_2 \circ f \circ {f_1}^{-1} \circ f_1\).
