44.4© 2012 Pearson Education, Inc.Vector SpacesCOORDINATE SYSTEMSSlide 4.4- 2© 2012 Pearson Education, Inc.THE UNIQUE REPRESENTATION THEOREM Theorem 7: Let  be a basis for vector space V. Then for each x in V, there exists a unique set of scalars c1, …, cnsuch that----(1) Proof: Since  spans V, there exist scalars such that (1) holds.  Suppose x also has the representationfor scalars d1, …, dn. 1{b ,...,b}n11xb...bnncc11xb...bnnddSlide 4.4- 3© 2012 Pearson Education, Inc.THE UNIQUE REPRESENTATION THEOREM Then, subtracting, we have----(2) Since  is linearly independent, the weights in (2) must all be zero. That is, for . Definition: Suppose  is a basis for Vand x is in V. The coordinates of x relative to the basis  (or the -coordinate of x) are the weights c1, …, cnsuch that .1110xx( )b...( )bnnncd cd   jjcd1 jn1{b ,...,b}n11xb...bnnccSlide 4.4- 4© 2012 Pearson Education, Inc.THE UNIQUE REPRESENTATION THEOREM If c1, …, cnare the -coordinates of x, then the vector in[x]is the coordinate vector of x (relative to ), or the -coordinate vector of x.  The mapping is the coordinate mapping(determined by ).1nccnxxSlide 4.4- 5© 2012 Pearson Education, Inc.COORDINATES IN  When a basis  for is fixed, the -coordinate vector of a specified x is easily found, as in the example below. Example 1: Let , , , and . Find the coordinate vector [x]of xrelative to . Solution: The -coordinate c1, c2of x satisfynn12b121b14x512{b,b}12214115ccb1b2xSlide 4.4- 6© 2012 Pearson Education, Inc.COORDINATES INor----(3) This equation can be solved by row operations on an augmented matrix or by using the inverse of the matrix on the left. In any case, the solution is , . Thus and .n1221 411 5cc   b1b2x13c22c12x3b 2b123x2BccSlide 4.4- 7© 2012 Pearson Education, Inc.COORDINATES IN See the following figure. The matrix in (3) changes the -coordinates of a vector x into the standard coordinates for x. An analogous change of coordinates can be carriedout in for a basis  .  Let Pnn1{b ,...,b}n12bbbnSlide 4.4- 8© 2012 Pearson Education, Inc.COORDINATES IN Then the vector equationis equivalent to----(4) Pis called the change-of-coordinates matrixfrom  to the standard basis in .  Left-multiplication by Ptransforms the coordinate vector [x]into x. Since the columns of Pform a basis for , Pis invertible (by the Invertible Matrix Theorem).n11 2 2xb b...bnncc cxxPBBnnSlide 4.4- 9© 2012 Pearson Education, Inc.COORDINATES IN Left-multiplication by converts x into its -coordinate vector:  The correspondence , produced by , is the coordinate mapping.  Since is an invertible matrix, the coordinate mapping is a one-to-one linear transformation from onto , by the Invertible Matrix Theorem.n1PB1xxPBBxx1PB1PBnnSlide 4.4- 10© 2012 Pearson Education, Inc.THE COORDINATE MAPPING Theorem 8: Let  be a basis for a vector space V. Then the coordinate mapping is a one-to-one linear transformation from V onto .  Proof: Take two typical vectors in V, say, Then, using vector operations, 1{b ,...,b}nxxn1111ub...bw b ... bnnnnccdd111u v ( )b ... ( )bnnncd cd   Slide 4.4- 11© 2012 Pearson Education, Inc.THE COORDINATE MAPPING It follows that  So the coordinate mapping preserves addition. If r is any scalar, then11 1 1uw u wnn n ncd c dcd c d  BBB11 1 1u(b...b)()b...()bnn n nrrc c rc rc Slide 4.4- 12© 2012 Pearson Education, Inc.THE COORDINATE MAPPING So Thus the coordinate mapping also preserves scalar multiplication and hence is a linear transformation. The linearity of the coordinate mapping extends to linear combinations. If u1, …, upare in V and if c1, …, cp are scalars, then----(5) 11uunnrc crrrrc c     BB11 1 1u ... u u ... upp p pccc c    BBBSlide 4.4- 13© 2012 Pearson Education, Inc.THE COORDINATE MAPPING In words, (5) says that the -coordinate vector of a linear combination of u1, …, upis the same linear combination of their coordinate vectors. The coordinate mapping in Theorem 8 is an important example of an isomorphism from V onto . In general, a one-to-one linear transformation from a vector space V onto a vector space W is called an isomorphism from V onto W. The notation and terminology for V and W may differ, but the two spaces are indistinguishable as vector spaces.nSlide 4.4- 14© 2012 Pearson Education, Inc.THE COORDINATE MAPPING Every vector space calculation in V is accurately reproduced in W, and vice versa. In particular, any real vector space with a basis of nvectors is indistinguishable from . Example 2: Let , , , and  . Then  is a basis for. Determine if x is in H, and if it is,find the coordinate vector of x relative to . n13v6221v013x12712{v,v}12Span{v,v}HSlide 4.4- 15© 2012 Pearson Education, Inc.THE COORDINATE MAPPING Solution: If x is in H, then the following vector equation is consistent: The scalars c1and c2, if they exist, are the -coordinates of x.123136012217ccSlide 4.4- 16© 2012 Pearson Education, Inc.THE COORDINATE MAPPING Using row operations, we