C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes4-7

# C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes4-7 -...

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SECTION 4.7 CHANGE OF BASIS Calculations in physics and engineering problems sometimes become simpler when we make good choices for the coordinate system, that is, the location of the origin and the direction of the axes we use when we translate the problem into mathematics. In this section we look at the effect of changing the axes. In our terminology, this means we want to change from one basis for a vector space to another basis for the same vector space. How does this change the coordinates of a given vector? A TWO-DIMENSIONAL EXAMPLE. Now suppose we have a basis B for an n -dimensional vector space V . Each vector x in V then has a B -coordinate vector [ x ] B . Then we have another basis C , so each vector x in V then has a C -coordinate vector [ x ] C . How do we get from one of these coordinate vectors to the other? Suppose B = { b 1 , b 2 , b 3 } and [ x ] B = r 1 r 2 r 3 . Now let’s calculate [ x ] C . FACT. To get [ x ] C from [ x ] B , multiply on the left by the change-of-coordinates matrix P C←B

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