Extra Example: Linear mapping from ‘# to ‘$ represented by matrices with respect to two
different bases
In class we looked at a linear mapping X À ‘8 Ä ‘8 where X ÐBÑ œ EB and talked about how
the matrix that represents X changes if we convert our calculations to coordinates from a bew
basis U (for both domain and codomain). Here's another example, where the domain and
codomain have different dimensions.
 B 
B
#
$
Example Suppose X À ‘ Ä ‘ is given by X Ð Ñ œ B  C .
C
B  C
The standard matrix for X is E œ Ò X Ð/" Ñ X Ð/# Ñ Ó œ
"
"
"
!
" Þ
"
When everything is written in standard coordinatesß X ÐBÑ œ EB
For example,
the point T
with standard
coordinates
"
#
its image in ‘$
in standard
coordinates is
 "
"
$
E  œ
#
 "
È
È
Choose new bases for domain and codomain:
For ‘#
U œ Ö," ß ,# ×
"
"
œ Ö ß 
×
#
"
For ‘$
V œ Ö-" ß -# ß -$ ×
! !  "
œÖ ! ß # ß " ×
" "  !
The change of coordinates can be diagrammed as follows:
mult by E
Ä
‘#
Æ Ò ÓU
‘#
‘$
Æ Ò ÓV
mult by Q
Ä
‘$
From the formula derived in text/lecture, the matrix of X with respect to the bases U , V is
Q œ Ò ÒX Ð," ÑÓV ÒX Ð,# ÑÓV Ó
Actually finding the matrix Q takes some work:
 -" 
To find ÒX Ð," ÑÓV œ -# ß we need to find the weights -" ß -# ß -$ so that
 -# 

"
X Ð," Ñ œ X Ð Ñ œ
#

so we need to solve
!
!
"
"
!
!
 "
$ œ -" !  -# #  -$  " :
"
"
 !
"
!
#
"
"  -"   " 
"
-# œ
$ Þ



!
-$
"
 ." 
Similarly, to find ÒX Ð,# ÑÓV œ .# we need to find the weights ." ß .# ß .$ so that
 .# 
X Ð,# Ñ œ X Ð
 "
!
!
 "
"
! œ ." !  .# #  .$  " :
ќ

"
 #
"
"
 !
!
so we need to solve !
"
!
#
"
"  ."    " 
"
.# œ
! Þ
!  .$    # 
We solve both systems at the same time by writing down both columns of constants and row
reducing the “double-augmented” matrix:
!
!
"
!
#
"
"
"
!
"
$
"
"
"
! µ ÞÞÞ µ  !
#
!
!
"
!
!
!
"
$
#
"
 $# 
 "# ,
"
so
$
 #
 -" 
 $
 ." 
-# œ ÒX Ð," ÑÓV œ
# and .# œ ÒX Ð,# ÑÓV œ   "# Þ
 -$ 
 "
 .$ 
 "
 $
Therefore Q œ  #
 "
 $# 
 "# is the matrix for X with respect to bases U and V .
"
To follow how a particular point T is treated
X
T ß stnd. coordinates
mult by E  " 
$
È
 "
"
#
Æ Ò ÓU
T ß in U coordinates
"
!
X ÐT Ñ, stnd. coordinates
Æ Ò ÓV
mult by Q   $ 
#
È
 "
X ÐT Ñ, in U coordinates
Very important conceptual comments
X is a linear mapping that sends vectors T in ‘# to vectors X ÐT Ñ in ‘$ . Start by imagining T as
a point in the plane and its image X ÐT Ñ as a point in space with no coordinate axes established
(so T and X ÐT Ñ, for the moment, do not have coordinates; imagine that X is somehow defined
geometrically rather than by a “formula.” The idea is that T and X ÐT Ñ are what they are: these
geometric points will be the same no matter how we decide, in the future, to set up a coordinate
system to describe them.
Then, we introduce coordinates to name these points (or vectors). We might do that by using
coordinates with respect to the standard bases f œ Ö/" ß /# × in ‘# and Ö/" ß /# ,/$ × in ‘$ . (The
basis vectors establish the coordinate axes, and give you the positive direction and scale on each
axis.)
"
If T can be written as "/"  #/# , then ÒT Óf œ  , and since f is the standard basis, we
#
"
usually don't even mention f  we just write T œ  Þ
#
This “standard way of assigning coordinates” is so routine in algebra and elementary calculus that
it has become subconscious for us  and therefore thinking about it now feels strange. We are so
"
accustomed to writing something like “T œ  ” that we usually think that “the point T is the
#
"
same thing as the coordinate vector  .” In other words, we ignore the fact that there are many
#
other ways to name the same geometric point T by setting up a different set of coordinate
axes  that is, by using coordinates with respect to some other basis. For example, with the basis
"
"
U used earlier: ÒT ÓU œ   is a new name for the same point T for which ÒT Óf œ  
!
#
In many situations we are not changing bases and then it's perfectly harmless to let the
"
standard basis f go unmentioned and just write T œ  . But there are other times
#
when we need to be more careful.
The linear transformation X maps T to its image X ÐT ÑÞ We can describe how this mapping
works in many ways algebraically: for example, changing the basis changes the matrix
associated with the linear mapping X , so X is described by a different formula. But when we
change the basis, the “picture” doesn't change: geometric point T still maps to the geometric
point X ÐT Ñ. Only the coordinates that name the points change (and that changes the matrix that
connects the coordinates of T to the coordinates of X ÐT ÑÑ.
X
T ß in std coordinates
"
#
Æ Ò ÓU
T ß in U coordinates
"
!
mult by E  " 
$
È
 "
X ÐT Ñ, in std coordinates
Æ Ò ÓV
mult by Q   $ 
#
È
 "
X ÐT Ñ, in U coordinates