three_v.pdf - Three.V Change of Basis Linear Algebra Jim...

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Three.V Change of Basis Linear Algebra Jim Hefferon
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Changing representations of vectors
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Coordinates vary with the basis Consider this vector ~ v R 3 and bases for the space. ~ v = 1 2 3 E 3 = h 1 0 0 , 0 1 0 , 0 0 1 i B = h 1 1 0 , 0 1 1 , 1 0 1 i With respect to the different bases, the coordinates of ~ v are different. Rep E 3 ( ~ v ) = 1 2 3 Rep B ( ~ v ) = 0 2 1 In this section we will see how to convert the representation of a vector with respect to a first basis to its representation with respect to a second.
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Change of basis matrix Think of translating from Rep B ( ~ v ) to Rep D ( ~ v ) as holding the vector constant. This is the arrow diagram. V wrt B id y V wrt D (This diagram is vertical to fit with the ones in the next subsection.)
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Change of basis matrix Think of translating from Rep B ( ~ v ) to Rep D ( ~ v ) as holding the vector constant. This is the arrow diagram. V wrt B id y V wrt D (This diagram is vertical to fit with the ones in the next subsection.) 1.1 Definition The change of basis matrix for bases B, D V is the representation of the identity map id : V V with respect to those bases. Rep B,D ( id ) = . . . . . . Rep D ( ~ β 1 ) · · · Rep D ( ~ β n ) . . . . . .
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1.3 Lemma Left-multiplication by the change of basis matrix for B, D converts a representation with respect to B to one with respect to D . Conversely, if left-multiplication by a matrix changes bases M · Rep B ( ~ v ) = Rep D ( ~ v ) then M is a change of basis matrix. Proof The first sentence holds because matrix-vector multiplication represents a map application and so Rep B,D ( id ) · Rep B ( ~ v ) = Rep D ( id ( ~ v ) ) = Rep D ( ~ v ) for each ~ v . For the second sentence, with respect to B, D the matrix M represents a linear map whose action is to map each vector to itself, and is therefore the identity map. QED
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Example To change a representation of a member of P 2 from being with respect to B = h 1, 1 + x, 1 + x + x 2 i to being with respect to D = h x 2 - 1, x, x 2 + 1 i , Compute Rep B,D ( id ) . The identity map acting on the elements of B has no effect, and so represent those elements with respect to D . Rep D ( 1 ) = - 1/2 0 1/2 Rep D ( 1 + x ) = - 1/2 1 1/2 Rep D ( 1 + x + x 2 ) = 0 1 1 To get the change of basis matrix just concatenate. Rep B,D ( id ) = - 1/2 - 1/2 0 0 1 1 1/2 1/2 1 For example, we can translate the representation of ~ v = 2 - x + 3x 2 . Rep B ( ~ v ) = 3 - 4 3 Rep D ( ~ v ) = 1/2 - 1 5/2 = - 1/2 - 1/2 0 0 1 1 1/2 1/2 1 3 - 4 3
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1.5 Lemma A matrix changes bases if and only if it is nonsingular.
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