An important topic in linear algebra is "change of basis", i.e., what happens to the representation of individual vectors, and of linear transformations, when bases of vector spaces are changed.
See, for example, Chapter VII, Section 7 of <I>Algebra, Third Edition</I> by Mac Lane and Birkhoff.
The machinery developed there applies to finite-dimensional vector spaces.
We can best understand that machinery by first considering the one-dimensional case.
In that case, the machinery is directly applicable to the elementary topic of "dimensional analysis":
https://en.wikipedia.org/wiki/Dimensional_analysis
Let us illustrate 'dimensional analysis" with an example.
Start with an approximation:
\[ \boxed{ \begin{array} {ccl|l} \source{ (1 \text{ inch}) } & = & \target{ (2.54 \text{ cm}) } & \text{the familiar statement of the approximation} \\ & = & \target{ \black{2.54} \times (1 \text{ cm}) } & \text{factoring out the conversion factor} \\ \source b & = & \target{ \black P \times c } = \target{ \black P \hat c } & \text{abstracting} \\ & = & \target{ (\source b \target y) \times c } = (\source b \target y) \target{\hat c} = \source b \target{(y \hat c)} = \source b & \text{see below for an explanation} \\ \end{array} } \] \[ \begin{array} {} &&& {} \rlap{ \kern-3em \target{ \text{(centimeters)} } } \\ &&& \target\R \\ &&& \target{ \llap{ \hat c = \text{centimeters} } \Bigg \downarrow } & \searrow \rlap{ P\inv = \text{($\target{\text{centimeters}}$ to $\source{\text{inches}}$) conversion} } \\ \source{ \text{(inches)} } & \source\R & \source{ \xrightarrow [\kern6em] { \textstyle \hat b = \text{inches} } } & V & \source{ \xrightarrow [ {} \rlap{ \kern-3em \textstyle x = (\hat b)\inv = \hom b V - = \langle b,- \rangle = \text{inches} } ] {\kern6em} } & \source\R & \source{ \text{(inches)} } \\ && \llap{ P = \text{($\source{\text{inches}}$ to $\target{\text{centimeters}}$) conversion} } \searrow & \target{ \Bigg \downarrow \rlap{ y = (\hat c)\inv = \hom c V - = \langle c,- \rangle = \text{centimeters} } } \\ &&& \target\R \\ &&& {} \rlap{\kern-3em \target{ \text{(centimeters)} } } \\ \end{array} \]
The method of dimensional analysis uses this equality to convert $\source{\text{measurements in inches}}$ to $\target{\text{measurements in centimeters}}$, e.g.
\[ \boxed{ \source{ 5 \text{ inches} } \times { \target{ 2.54 \text{ cm} } \over \source{ 1 \text{ inch} } } = \source 5 \times \target{ 2.54 \text{ cm} } } . \]
We now make a prolonged segue into mathematics, eventually to see how the above can be viewed as a special case of some general mathematical considerations.
So let us now move into the realms of geometry and algebra.
First, in geometry, we might consider a line $L$, extending infinitely in both directions, with points $P,Q$ etc.
We might assign (a base point, call it $0$, on the line), and wish to measure off distances along the line. But what units to use? The choice is up in the air.
Next, in algebra, more specifically linear algebra,
we may view the line as a 1-dimensional real vector space, consisting of vectors $v,w$ etc.
Being a vector space means that certain operations are defined on its elements (vectors):
A nullary operation, $0$, selecting the origin of the vector space,
addition of vectors, yielding $v+w$,
and scalar multiplication by reals, yielding $v \xi$ (we here, following Mac Lane and Birkhoff, use Greek letters to denote scalars).
All of these may be defined geometrically, given a bijection between (the line $L$) and (the 1-dimensional real vector space $V$).
Note that while reals (scalars) can <I>act</I> on $V$: $\langle v,\xi \rangle \mapsto v\xi$,
we do not yet have a <i>bijection</i> between the 1-dimensional real vector spaces $V$ and $\R$, the vector space of the scalars themselves.
I.e., we have no sense of scale within $V$.
We get that by $\source{ \text{choosing one non-zero vector } \boxed{b} \text{ in $V$ to be the <i>unit vector</i> or <I>base</I>.} }$
Given (the vector $\source b$), we have a function denoted $\source{ \boxed{ \hat b } }$:
\[ \source{ \boxed{ \hat b : \R \to V : \xi \mapsto b\xi } } , \]
taking (a scalar $\xi$) (which we may regard as a coordinate relative to the unit, or base, vector $\source b$) to (the vector $\source b \xi$).
Since ($\source b$ is a non-zero vector in a 1-dimensional vector space), (the function $\source{ \hat b }$) is in fact a bijection.
We denote the inverse to $\source{ \hat b }$ by $\source{ \boxed{x} }$, so $\source{ x = (\hat b)\inv }$. Thus
\[ \source{ \boxed{ x = (\hat b)\inv : V \to \R } } ; \]
it takes a ($v\in V$) to (the unique $\source{x}(v) = \xi \in\R$ such that $v = \source b \xi = \source b \source x (v) = \source{\hat b x}(v) $),
which we naturally regard as (the coordinate of $v$) relative to (the base vector $\source b$).
Since $\source{\hat b}$ and $\source x$ are inverse to each other, we have $\source{\hat b x = 1_\R}$ and $\source{x \hat b} = 1_V$ (note the order of composition being used).
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Some diagrams for future use:
For an explanation of the following diagram, see Thm. 11 in M-B VI.4
\[ \begin{array} {} 1 \\ &&&& \searrow \rlap {v \in V} \\ & \llap{ n\op \backslash F \ni \mathbf\xi} \searrow & n \rlap{ \xrightarrow [\kern 11em] {\kern 1em \textstyle \mathbf b} } &&&& V \\ &&& \llap{ \hom - n - \lambda } \searrow && \llap{ \mathbf\mu } \searrow && \searrow \rlap{ f \in V^\star } \\ &&&& n\op \backslash F \rlap{ \kern.5em \xrightarrow [\textstyle \mathbf\mu \star -] {\kern 8em} } &&&& F \\ \end{array} \]
The short (1-2) diagonal (not shown) in the diagram is $\mathbf b \star -$), i.e. the extension of $\mathbf b$ over $\hom - n - \lambda$.
A special case of the the change of base diagram.
Here we consider a line,
with a given origin or base point,
as a one dimensional vector space,
and consider how it may be measured in either inches or centimeters.
\[ \begin{array} {} {} \rlap{ \source { m=1 } } &&&&&&& {} \rlap{ \target{ n=1} } \\ & \source{ \llap{\mathbf b} \searrow \rlap{\text{inch}} } &&&&&&& \kern-2em \target{ \llap{\mathbf c} \searrow \rlap{\text{centimeter}} } \\ && \source{ V \rlap{ \kern.5em \xrightarrow {\kern9em} } } &&&& \source V & \kern-.5em \xrightarrow{\kern6em} & \target{ V \rlap{ \kern.5em \xrightarrow {\kern9em} } } &&&& \target V \\ &&& \source{ \llap{ \mathbf x = \hom {\mathbf b} V - } \searrow } & \source \Vert &\source{ \nearrow \rlap{ \mathbf b \star - } } &&&& \target{ \llap{ \mathbf y = \hom {\mathbf c} V -} \searrow } & \target\Vert & \target{ \nearrow \rlap{ \mathbf c \star - } } \\ &&&& \source{ m\calP } \rlap{ \kern.5em \xrightarrow [ \textstyle (P = \hom {\target{\mathbf c}} {\target V} {\source{\mathbf b}} ) \star {\source -} ] {\kern18em} } &&&&&& \target{n\calP} \\ \end{array} \]
Several of these arrows represent mappings which have significant verbal descriptions.
The diagonal arrows, taking them from left to right, may be described verbally as:
Selecting the point corresponding to one inch ; selecting a unit (an inch).
The coordinate (in inches) of a point ; converting points to numbers (of inches); this is the act of <I>measurement</I>, in inches.
The point determined by a specific number of inches ; converting numbers (of inches) to points.
The next (and last) three diagonal arrows do the same for centimeters.
\[ \begin{array} {} \source n &&&&&&& {} \rlap{ \target n } \\ & \source{ \llap{\mathbf b} \searrow } &&&&&&& \kern-2em \target{ \searrow \rlap{\mathbf c} } \\ && \source{ V \rlap{ \kern.5em \xrightarrow {\kern9em} } } &&&& \source V & \kern-.5em \xrightarrow{\kern6em} & \target{ V \rlap{ \kern.5em \xrightarrow {\kern9em} } } &&&& \target V \\ &&& \source{ \llap{ \mathbf x = \hom {\mathbf b} V - } \searrow } & \source \Vert &\source{ \nearrow \rlap{ \mathbf b \star - } } &&&& \target{ \llap{ \mathbf y = \hom {\mathbf c} V -} \searrow } & \target\Vert & \target{ \nearrow \rlap{ \mathbf c \star - } } \\ &&&& \source{ n\calP } \rlap{ \kern.5em \xrightarrow [ \textstyle (P = \hom {\target{\mathbf c}} {\target V} {\source{\mathbf b}} ) \star {\source -} ] {\kern18em} } &&&&&& \target{n\calP} \\ \end{array} \]
The change of base diagram, modified in two ways.
First, for a linear transformation $t$ from a $m$-dimensional vector space $V$ to an $n$-dimensional vector space $W$.
Second, specializing to one dimensional vector spaces (so $\source{m=1}$ and $\target{n=1}$),
and assuming the bases for the two vector spaces are, respectively,
points at a distance of one inch and one centimeter from the origin.
\[ \begin{array} {} {} \rlap{ \source { m=1 } } &&&&&&& {} \rlap{ \target{ n=1} } \\ & \source{ \llap{\mathbf b} \searrow \rlap{\text{inch}} } &&&&&&& \kern-2em \target{ \llap{\mathbf c} \searrow \rlap{\text{centimeter}} } \\ && \source{ V \rlap{ \kern.5em \xrightarrow {\kern9em} } } &&&& \source V & \kern-.5em \xrightarrow[\kern8em] {\textstyle t} & \target{ W \rlap{ \kern.5em \xrightarrow {\kern9em} } } &&&& \target W \\ &&& \source{ \llap{ \mathbf x = \hom {\mathbf b} V - } \searrow } & \source \Vert &\source{ \nearrow \rlap{ \mathbf b \star - } } &&&& \target{ \llap{ \mathbf y = \hom {\mathbf c} W -} \searrow } & \target\Vert & \target{ \nearrow \rlap{ \mathbf c \star - } } \\ &&&& \source{ m\calP } \rlap{ \kern.5em \xrightarrow [ \textstyle (P = \hom {\target{\mathbf c}} {\target W} {\source{\mathbf b}t} ) \star {\source -} ] {\kern20em} } &&&&&& \target{n\calP} \\ \end{array} \]
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THIS IS DRAFT FOR A RATHER DIFFERENT SUBJECT
(Mac Lane-Birkhoff affine spaces a la Janelidze-Kelly).
\[ {\mathcal V}_0 (Y,Z) \cong {\mathcal V}_0 (I, [Y,Z]) \equiv V([Y,Z]) \tag*{ECT (1.25)} \]
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\[ \kappa : \hom {v+q} {P} p \buildrel \kappa \over \cong \hom v {V} { ( \hom q {\mathbf P} p = (p-q) ) } \tag{MB 5, JK 2.1} \]
