Linear Algebra, 2020a

186 Chapter Three. Maps Between Spaces
This function is one-to-one because if
Rep B (u 1
⃗β 1 + ···+ u n
⃗β n )=Rep B (v 1
⃗β 1 + ···+ v n
⃗β n )
then
⎛ ⎞
u 1
⎜
⎝ .
u n
⎟
⎠ =
⎛ ⎞
v 1
⎜ ⎟
⎝ . ⎠
v n
and so u 1 = v 1 , ..., u n = v n , implying that the original arguments u 1
⃗β 1 +
···+ u n
⃗β n and v 1
⃗β 1 + ···+ v n
⃗β n are equal.
This function is onto; any member of R n
⃗w =
⎛ ⎞
w 1
⎜ .
⎝
⎟
. ⎠
w n
is the image of some ⃗v ∈ V, namely ⃗w = Rep B (w 1
⃗β 1 + ···+ w n
⃗β n ).
Finally, this function preserves structure.
Rep B (r · ⃗u + s · ⃗v) =Rep B ((ru 1 + sv 1 )⃗β 1 + ···+(ru n + sv n )⃗β n )
⎛ ⎞
ru 1 + sv 1
⎜
= ⎝
⎟
. ⎠
ru n + sv n
= r ·
⎛
⎜
⎝
⎞
u 1
. .
u n
⎟
⎠ + s ·
⎛
⎜
⎝
⎞
v 1
. ⎟
. ⎠
v n
= r · Rep B (⃗u)+s · Rep B (⃗v)
Therefore Rep B is an isomorphism. Consequently any n-dimensional space
is isomorphic to R n .
QED
2.6 Remark When we introduced the Rep B notation for vectors on page 125, we
noted that it is not standard and said that one advantage it has is that it is
harder to overlook. Here we see its other advantage: this notation makes explicit
that Rep B is a function from V to R n .
2.7 Remark The proof has a sentence about ‘well-defined.’ Its point is that to be
an isomorphism Rep B must be a function. The definition of function requires
that for all inputs the associated output must exists and must be determined by
the input. So we must check that every ⃗v is associated with at least one Rep B (⃗v),
and with no more than one.