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Figure FTMR, discussion needs a go (Jake McRae)
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rbeezer committed Apr 24, 2021
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Expand Up @@ -191,6 +191,7 @@ Typo: Sage GS, remove a stray comment (Chrissy Safranski)
Typo: Sage LNS, unbalanced parenthesis (Maggie Hatt)
Typo: Solution EE.M60, correct and clarify eigenvectors (Anna Van Boven)
Typo: Theorem SMEE, roles of A and B reversed (Jake Fischer)
Typo: Figure FTMR, discussion needs a "go" (Jake McRae)

v3.50 2015/12/30
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2 changes: 1 addition & 1 deletion src2/section-MR.xml
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</latex-image>
</image>
</figure>
<p>The alternative conclusion of this result might be even more striking. It says that to effect a linear transformation (<m>T</m>) of a vector (<m>\vect{u}</m>), coordinatize the input (with <m>\vectrepname{B}</m>), do a matrix-vector product (with <m>\matrixrep{T}{B}{C}</m>), and un-coordinatize the result (with <m>\vectrepinvname{C}</m>). So, absent some bookkeeping about vector representations, a linear transformation <em>is</em> a matrix. To adjust the diagram, we <q>reverse</q> the arrow on the right, which means inverting the vector representation <m>\vectrepname{C}</m> on <m>V</m>. Now we can go directly across the top of the diagram, computing the linear transformation between the abstract vector spaces. Or, we can around the other three sides, using vector representation, a matrix-vector product, followed by un-coordinatization.</p>
<p>The alternative conclusion of this result might be even more striking. It says that to effect a linear transformation (<m>T</m>) of a vector (<m>\vect{u}</m>), coordinatize the input (with <m>\vectrepname{B}</m>), do a matrix-vector product (with <m>\matrixrep{T}{B}{C}</m>), and un-coordinatize the result (with <m>\vectrepinvname{C}</m>). So, absent some bookkeeping about vector representations, a linear transformation <em>is</em> a matrix. To adjust the diagram, we <q>reverse</q> the arrow on the right, which means inverting the vector representation <m>\vectrepname{C}</m> on <m>V</m>. Now we can go directly across the top of the diagram, computing the linear transformation between the abstract vector spaces. Or, we can go around the other three sides, using vector representation, a matrix-vector product, followed by un-coordinatization.</p>
<figure xml:id="figure-FTMRA" acro="FTMRA">
<caption>Fundamental Theorem of Matrix Representations (Alternate)</caption>
<image xml:id="image-FTMRA">
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