Another Example of Changing Coordinates Rotating the Coordinate System

cos9
sin 9 Suppose 9 is some specific fixed angle. The matrix is invertible ( what is ”sin 9 cos 9 • its determinant? ), so its columns are linearly independent and span ‘# . Therefore

cos9
sin 9 U œÖ,,ß ×œÖ ß × is a basis for ‘ # . The vectors , and , determine a "# ”sin 9 •” cos 9 • " #

new set of coordinate axes (as pictured below) which, for convenience, we will call the Bw and C w w w w B axes
so that U-coordinates are the same thing as B - C coordinates: Ò BÓU œ . ”Cw •

U-coordinates represent measurements along the Bw and C w axes (where, for example, the “tip” of B the vector , is “one unit” in the positive direction along the Bw axis). Standard coordinates " ”C • are measured along the BC - axes (determined by the standard basis vectors /" and / # ÑÞ

The formulas for converting between U -coordinates and standard coordinates are

cos 9
sin 9 Bw B T ÒB Ó œ B , that is, œ Ð‡Ñ U U ”sin9 cos 9 •”•”•Cw C Å Å ," , #

cos 9sin 9 B B w ÒÓœTB
" B , that is, œ Ї‡Ñ U U ”
sin 9cos 9 •”•”• C C w You may have seen these “rotation of axes” formula written out, in precalculus or calculus, in a harder to remember form:

BœBw cos99
Cw sin BœBw cos 99 C sin Ð‡Ñ and Ї‡Ñ œ C œ Bw sin 99  Cw cosœ Cw œ
B sin 99  C cos

In the future, to get these formulas you just need to remember how to write a rotation matrix and the formula TU ÒB ÓU œ B .

Here's an example where changing coordinate systems (specifically, rotating axes) gives some new insight. What does the graph of &B#
%BC&C # œ'$ look like ?

The expression &B#
%BC &C # is an example of something called a “quadratic form” and we may have time, at the very end of the course, to look briefly at the general theory of quadratic forms.

In any case, the general theory suggests that, for this example, a rotation of coordinates through 1 the angle9 œ % would be useful. Substituting according to equations Ї‡Ñ , we convert into Bw- C w coordinates.

'$ œ &B#
%BC&C #

œ&ÐBwcos99
Cw sin Ñ
%ÐB #w cos 9999
C ww sin ÑÐB wsin C cos Ñ&ÐB w sin 99 C w cos Ñ #

1 With 9 œ % , we get

wÈÈÈÈÈÈÈÈ##w #w #### ww w w ## w # &ÐB##
C Ñ
%ÐB ####
C ÑÐB C Ñ&ÐB ## C Ñœ'$

Multiplying out and simplifying gives:

"""w# ww w# "" w# w# """ w# ww w# &Ð#%# ÐBÑ
BC ÐCÑÑ
%Ð ## ÐBÑ
ÐCÑ Ñ&Ð #%# ÐBÑ  BC ÐCÑÑœ'$ which simplifies to $ÐBÑw# (ÐCÑ w# œ'$ , or

ÐBÑw# ÐCÑ w# or #" * œ " Þ

In BCw - w coordinates, there is no “mixed” middle term Bw C w ß and we can recognize the graph as being an ellipse (whose axes are the BCw and w axes). With respect to the original BC and axes, the ellipse is “tilted” which makes its equation more complicated when we work in the original B- C coordinates.

ÐSee the graph on the next page. Ñ

ÐBÑw# ÐCÑ w# #" * œ " Þ

&B#
%BC&C # œ'$

An Important Observation: Two Points of View

Point of View 1 )

Consider a rotation of ‘# counterclockwise 90° around the origin (that is, through a positive 1 # angle # œ *! °). This transformation of ‘ is performed by a rotation matrix ( see Chapter 2 ):

1 1 cos#
sin # !
" EœÒXÐÑXÐÑÓœ/" / # 1 1 œ ”sin# cos # •”" ! •

We picture “multiplying by E ” as a moving points in ‘# to new locations:

B B B È E œ “the result of rotating counterclockwise ”•C ”• C ”• C 1 around the origin by # œ 90° !
" " ! For example E/œ œ œ /œ/ “ rotated 90° " ”" ! • ”•! ”• " # " counterclockwise ”

Of course, everything here is written in “standard coordinates”
back in Chapter 2, the idea of # using a new coordinate system ÐÒÓ coordinates U from a different basis for ‘ Ñ hadn't even been mentioned.

Point of View 2 ) In Chapter 4, we now consider the idea of changing coordinates
for example, by rotating coordinate axes .

!
" Suppose we take a new basis for ‘# , say U œÖßלÖ,, ß × and let ,, ß "# ”•”" ! • " # w w determine the new coordinate axes: the new axis B runs through ," and the new C axis runs w w through ,#Þ Let's refer to the U -coordinates of a point as its B C coordinates.

!
" The matrix TœÒ, , Óœ is then the “change of coordinate matrix” U " # ”" ! •

from U -coordinates to standard coordinates: TU ÒB Ó U œ B , that is

!
"Bw B !
"" ! œ . For example œÐ‡‡Ñ ”"!C •”•”•w C ” "!!" •”•”•

Here, we are not thinking of “moving” points at all. A point T (to give it a neutral name ) Bw 1 described in U coordinates might be œ ”Cw • ”•0

" !
"" ! B Then multiplication Tœ œœ merely computes the U”•! ” "!! •”• ”• " ”• C standard coordinates of the same point T À no points have moved ; instead, the coordinate axes have been moved . !
" " ! So: what does the matrix equation œ mean , geometrically? ”" ! • ”•! ”• "

The answer is that it can have at least two different interpretations:

" ! i) a transformation that“moves” to the point (by rotating *! °) ”•! ”• "

or ii) a change of the coordinates of a fixed point T ß

B"w B! from œ to œ ”•”•C!w ”•”• C"

!
" " ! When you see œ , you have to keep in mind how to interpret it in the ”" ! • ”•! ”• " particular situation. .

This kind of thing is not at all surprising in mathematics: that the very same equation can be interpreted to mean different things. In fact, that's one reason why mathematics is so useful: the same tool or equation can have many different interpretations. For example, in calculus: what does " # mean ? '! > .>

" # " can mean “the area under the graph of # and above the interval '! > .> œ $ Cœ> Ò!ß"Ó

" # " can mean “the amount of work done by a variable force # applied '! > .> œ $ JÐ>Ñ œ > at > to move an object from !"> to along the -axis”

" # " can mean “the volume of the solid obtained when the region above '! > .> œ $ Ò!ß "Ó and under the graph of Cœ0Ð>Ñœ" > is revolved around È 1 the >-axis” (recall that the volume of this solid of revolution can becomputed as "1Ð0Ð>ÑÑ.>œ# " 1 Ð" >Ñ.> # '! ' ! È 1 " # . œ'! >.>