Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
4 Vector Spaces 4.4 COORDINATE SYSTEMS © 2012 Pearson Education, Inc. THE UNIQUE REPRESENTATION THEOREM Theorem 7: Let B {b1 ,...,bn } be a basis for vector space V. Then for each x in V, there exists a unique set of scalars c1, …, cn such that ----(1) x c1b1 ... cn bn Proof: Since B spans V, there exist scalars such that (1) holds. Suppose x also has the representation x d1b1 ... dn bn for scalars d1, …, dn. © 2012 Pearson Education, Inc. Slide 4.4- 2 THE UNIQUE REPRESENTATION THEOREM Then, subtracting, we have 0 x x (c1 d1 )b1 ... (cn d n )b n ----(2) Since B is linearly independent, the weights in (2) must all be zero. That is, c j d j for 1 j n . Definition: Suppose B {b1 ,...,bn } is a basis for V and x is in V. The coordinates of x relative to the basis B (or the B-coordinate of x) are the weights c1, …, cn such that x c1b1 ... cn b n . © 2012 Pearson Education, Inc. Slide 4.4- 3 THE UNIQUE REPRESENTATION THEOREM If c1, …, cn are the B-coordinates of x, then the vector n c1 in [x]B cn is the coordinate vector of x (relative to B), or the B-coordinate vector of x. The mapping x x B is the coordinate mapping (determined by B). © 2012 Pearson Education, Inc. Slide 4.4- 4 COORDINATES IN n When a basis B for is fixed, the B-coordinate vector of a specified x is easily found, as in the example below. 2 1 4 Example 1: Let b1 , b 2 , x , and n 1 1 5 B {b1 ,b 2 }. Find the coordinate vector [x]B of x relative to B. Solution: The B-coordinate c1, c2 of x satisfy 2 1 4 c1 c2 1 1 5 © 2012 Pearson Education, Inc. b1 b2 x Slide 4.4- 5 COORDINATES IN or n 2 1 c1 4 1 1 c 5 2 b1 b2 ----(3) x 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 c1 3 , c2 2 . Thus x 3b1 2b 2 and c1 3 . x © 2012 Pearson Education, Inc. B c2 2 Slide 4.4- 6 COORDINATES IN n See the following figure. The matrix in (3) changes the B-coordinates of a vector x into the standard coordinates for x. An analogous change of coordinates can be carried n out in for a basis B {b1 ,...,bn }. Let PB b1 b2 © 2012 Pearson Education, Inc. bn Slide 4.4- 7 COORDINATES IN n Then the vector equation is equivalent to x c1b1 c2 b2 ... cn bn x PB x B ----(4) PB is called the change-of-coordinates matrix n from B to the standard basis in . Left-multiplication by PB transforms the coordinate vector [x]B into x. Since the columns of PB form a basis for , PB is invertible (by the Invertible Matrix Theorem). n © 2012 Pearson Education, Inc. Slide 4.4- 8 COORDINATES IN n 1 B Left-multiplication by P converts x into its Bcoordinate vector: P x x B 1 B The correspondence x 1 x B, produced by PB , is the coordinate mapping. 1 Since PB is an invertible matrix, the coordinate mapping is a one-to-one linear transformation from n onto , by the Invertible Matrix Theorem. © 2012 Pearson Education, Inc. n Slide 4.4- 9 THE COORDINATE MAPPING Theorem 8: Let B {b1 ,...,bn } be a basis for a vector x nB is a space V. Then the coordinate mapping x one-to-one linear transformation from V onto . Proof: Take two typical vectors in V, say, u c1b1 ... cn b n w d1b1 ... d n b n Then, using vector operations, u v (c1 d1 )b1 ... (cn dn )b n © 2012 Pearson Education, Inc. Slide 4.4- 10 THE COORDINATE MAPPING It follows that u w B c1 d1 c1 d1 u w B B cn d n cn d n So the coordinate mapping preserves addition. If r is any scalar, then ru r (c1b1 ... cn bn ) (rc1 )b1 ... (rcn )b n © 2012 Pearson Education, Inc. Slide 4.4- 11 THE COORDINATE MAPPING So ru B rc1 c1 r r u B rcn cn 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, …, up are in V and if c1, …, cp are scalars, then c1u1 ... c p u p c1 u1 B ... c p u p ----(5) B © 2012 Pearson Education, Inc. B Slide 4.4- 12 THE COORDINATE MAPPING In words, (5) says that the B-coordinate vector of a linear combination of u1, …, up is the same linear combination of their coordinate vectors. The coordinate mapping in Theorem 8 is an important n 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. © 2012 Pearson Education, Inc. Slide 4.4- 13 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 n vectors is indistinguishable from n . 3 1 3 Example 2: Let v1 6 , v 2 0 , x 12 , 2 1 7 and B {v1 , v2 }. Then B is a basis for H Span{v1 ,v2 } . Determine if x is in H, and if it is, find the coordinate vector of x relative to B. © 2012 Pearson Education, Inc. Slide 4.4- 14 THE COORDINATE MAPPING Solution: If x is in H, then the following vector equation is consistent: 3 1 3 c1 6 c2 0 12 2 1 7 The scalars c1 and c2, if they exist, are the Bcoordinates of x. © 2012 Pearson Education, Inc. Slide 4.4- 15 THE COORDINATE MAPPING Using row operations, we obtain 3 1 3 6 0 12 2 1 7 1 0 2 0 1 3 . 0 0 0 2 Thus c1 2 , c2 3 and [x]B . 3 © 2012 Pearson Education, Inc. Slide 4.4- 16 THE COORDINATE MAPPING The coordinate system on H determined by B is shown in the following figure. © 2012 Pearson Education, Inc. Slide 4.4- 17