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571_3.5 - 3.5 CHANGE OF BASIS KIAM HEONG KWA p 150 Let E =...

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3.5 CHANGE OF BASIS KIAM HEONG KWA p. 150 Let E = { v 1 , v 2 , · · · , v n } be an ordered basis for a finite dimen- sional linear space V . The coordinate vector of an element v V with respect to the ordered basis E is the unique vector c = ( c 1 , c 2 , · · · , c n ) T R n such that v = n X i =1 c i v k = c 1 v 1 + c 2 v 2 + · · · + c n v n . The coordinates c i ’s of c are called the coordinates of v with re- spect to E . Notation: Such a vector c is denoted [ v ] E . p. 151 Let V be a linear space of finite dimension n . A change from an ordered basis E = { w 1 , w 2 , · · · , w n } of V to another F = { v 1 , v 2 , · · · , v n } effects a corresponding change in the coordinate systems. To see this, let v V but otherwise arbitrary and let [ v ] E = ( x 1 , x 2 , · · · , x n ) T and [ v ] F = ( y 1 , y 2 , · · · , y n ) T . Let s ij ’s be the n 2 scalars such that w j = n X i =1 s ij v i = s 1 j v 1 + s 2 j v 2 + · · · + s nj v n for each j ∈ { 1 , 2 , · · · , n } . If S = ( s ij ) is the n × n matrix whose ( i, j ) entry is s ij , then the above equations can be written compactly as the matrix equation ( w 1 w 2 · · · w n ) = ( v 1 v 2 · · · v n ) S, where we have treated ( w 1 w 2 · · · w n ) and ( v 1 v 2 · · · v n ) as matrices whose entries are vectors. Note tha if V is the n -dimensional Euclidean space R n , then S = ( v 1 v 2 · · · v n ) - 1 ( w 1 w 2 · · · w n ) .
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