Another Example of hanging oordinates Rotating the oordinate System cos9 sin 9 Suppose 9 is some specific fixed angle. The matrix sin 9 cos9 is invertible (hat is its determinant?), so its columns are linearly independent and span #. Therefore cos U 9 sin œ Ö œ Ö ß 9 #, ß,#,,# sin 9 cos9 is a basis for. The vectors and determine a ne set of coordinate axes (as pictured belo) hich, for convenience, e ill call the and axes so that U-coordinates are the same thing as - coordinates: ÒÓ U œ. U-coordinates represent measurements along the and axes (here, for example, the tip of the vector, is one unit in the positive direction along the axis). Standard coordinates are measured along the - axes (determined by the standard basis vectors / and / ÑÞ # The formulas for converting beteen U-coordinates and standard coordinates are T Ò Ó œ cos9 sin9, that is, sin U U 9 cos9 Ð Ñ Å Å,,# sin ÒÓU œ TU cos9 9, that is, œ Ð Ñ sin9 cos9
You may have seen these rotation of axes formula ritten out, in precalculus or calculus, in a harder to remember form: œ cos9 sin9 œ cos9 sin9 Ð Ñœ and Ð Ñ œ sin œ 9 cos 9 œ sin9 cos9 In the future, to get these formulas you just need to remember ho to rite a rotation matrix and the formula T Ò Ó œ U U. Here's an example here changing coordinate systems (specifically, rotating axes) gives some ne insight. What does the graph of & % & œ '$ look like? The expression & % & is an example of something called a quadratic form and e 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 angle 9 œ ould be useful. Substituting according to equations Ð Ñ, e convert into % - coordinates. '$ œ & % & œ &Ð cos9 sin9ñ %Ð cos9 sin 9ÑÐ sin9 cos 9Ñ &Ð sin 9 cos9ñ With # #, e get 9 œ 1 % È # È # È # È # È # È # È # È # # # &Ð Ñ %Ð ÑÐ Ñ &Ð Ñ œ '$ Multiplying out and simplifying gives: # % % # # # # # # # &Ð Ð Ñ Ð Ñ Ñ %Ð Ð Ñ Ð Ñ Ñ &Ð Ð Ñ Ð Ñ Ñ œ '$ hich simplifies to # # $Ð Ñ (Ð Ñ œ '$, or # # or Ð Ñ Ð Ñ # * œ Þ In - coordinates, there is no mixed middle term ß and e can recognize the graph as being an ellipse (hose axes are the and axes). With respect to the original and axes, the ellipse is tilted hich makes its equation more complicated hen e ork in the original - coordinates. ÐSee the graph on the next page. Ñ
# # Ð Ñ Ð Ñ # * œ Þ & % & œ '$ An Important Observation: To Points of Vie Point of Vie 1) onsider a rotation of # counterclockise 90 around the origin (that is, through a positive 1 # angle œ * ). This transformation of is performed by a rotation matrix ( see hapter 2): # 1 1 cos sin E œ ÒXÐ/ Ñ XÐ/ # ÑÓ œ 1 1 œ sin cos We picture multiplying by E as a moving points in # to ne locations: È E œ the result of rotating counterclockise 1 around the origin by 90 # œ
For example E/ œ œ œ /# œ / rotated 90 counterclockise Of course, everything here is ritten in standard coordinates back in hapter 2, the idea of using a ne coordinate system Ðcoordinates ÒÓU from a different basis for # Ñ hadn't even been mentioned. Point of Vie 2) In hapter 4, e no consider the idea of changing coordinates for example, by rotating coordinate axes. # Suppose e take a ne basis for, say U œ Ö, ß,# œ Ö ß and let, ß, # determine the ne coordinate axes: the ne axis runs through, and the ne axis runs through, # Þ Let's refer to the U-coordinates of a point as its coordinates. The matrix T œ Ò, Ó œ is then the change of coordinate matrix U,# fromu-coordinates to standard coordinates: T ÒÓ œ, that is. For example œ œ Ð Ñ Here, e are not thinking of moving points at all. A point T ( to give it a neutral name) described in U coordinates might be œ 1 0 U U Then multiplication TU œ œ œ merely computes the standard coordinates of the same point T À no points have moved; instead, have been moved. the coordinate axes
So: hat does the matrix equation œ mean, geometrically? The anser is that it can have at least to different interpretations: i) a transformation that moves to the point (by rotating * ) or ii) a change of the coordinates of a fixed point Tß from to œ œ When you see, you have to keep in mind ho to interpret it in the œ 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 hy mathematics is so useful: the same tool or equation can have many different interpretations. For example, in calculus: hat does ' > #.> mean? ' # # can mean area $ >.> œ the under the graph of œ > and above the interval ÒßÓ ' # # >.> œ can mean ork JÐ>Ñ œ > $ ' # $ the amount of done by a variable force applied at > to move an object from to along the > -axis >.> œ can mean the volume of the solid obtained hen the region above ÒßÓ and under the graph of œ 0Ð>Ñ œ > is revolved around the > -axis (recall that the volume of this solid of revolution can be computed as ' # ' # 1Ð0Ð>ÑÑ.> œ 1Ð >Ñ.> œ ' > #.>. È1 È1