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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 © 2012 Pearson Education, Inc. Slide 4.4- 2 THE UNIQUE REPRESENTATION THEOREM 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 . © 2012 Pearson Education, Inc. Slide 4.4- 10 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- 11 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- 12 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- 13 THE COORDINATE MAPPING The coordinate system on H determined by B is shown in the following figure. © 2012 Pearson Education, Inc. Slide 4.4- 14 © 2012 Pearson Education, Inc. Slide 4.4- 15 4 Vector Spaces 4.5 THE DIMENSION OF A VECTOR SPACE © 2012 Pearson Education, Inc. DIMENSION OF A VECTOR SPACE Theorem 9: If a vector space V has a basis B {b1 ,...,bn }, then any set in V containing more than n vectors must be linearly dependent. Theorem 9 implies that if a vector space V has a basis B {b1 ,...,bn }, then each linearly independent set in V has no more than n vectors. © 2012 Pearson Education, Inc. Slide 4.5- 17 DIMENSION OF A VECTOR SPACE Definition: If V is spanned by a finite set, then V is said to be finite-dimensional, and the dimension of V, written as dim V, is the number of vectors in a basis for V. The dimension of the zero vector space {0} is defined to be zero. If V is not spanned by a finite set, then V is said to be infinite-dimensional. Example 1: Find the dimension of the subspace a 3b 6c 5 a 4 d : a, b, c, d in H b 2c d 5d © 2012 Pearson Education, Inc. Slide 4.5- 18 DIMENSION OF A VECTOR SPACE H is the set of all linear combinations of the vectors 1 3 6 0 5 0 0 4 v1 , v 2 , v3 , v 4 0 1 2 1 0 0 0 5 Clearly,v1 0 , v2 is not a multiple of v1, but v3 is a multiple of v2. By the Spanning Set Theorem, we may discard v3 and still have a set that spans H. © 2012 Pearson Education, Inc. Slide 4.5- 19 SUBSPACES OF A FINITE-DIMENSIONAL SPACE Finally, v4 is not a linear combination of v1 and v2. So {v1, v2, v4} is linearly independent and hence is a basis for H. Thus dim H 3. Theorem 11: Let H be a subspace of a finitedimensional vector space V. Any linearly independent set in H can be expanded, if necessary, to a basis for H. Also, H is finite-dimensional and dim H dimV © 2012 Pearson Education, Inc. Slide 4.5- 20 THE BASIS THEOREM Theorem 12: Let V be a p-dimensional vector space, p 1 . Any linearly independent set of exactly p elements in V is automatically a basis for V. Any set of exactly p elements that spans V is automatically a basis for V. © 2012 Pearson Education, Inc. Slide 4.5- 21 DIMENSIONS OF NUL A AND COL A Thus, the dimension of Nul A is the number of free variables in the equation Ax 0 , and the dimension of Col A is the number of pivot columns in A. Example 2: Find the dimensions of the null space and the column space of 3 6 1 1 7 A 1 2 2 3 1 2 4 5 8 4 © 2012 Pearson Education, Inc. Slide 4.5- 22 DIMENSIONS OF NUL A AND COL A Solution: Row reduce the augmented matrix A 0 to echelon form: 1 2 2 3 1 0 0 0 1 2 2 0 0 0 0 0 0 0 There are three free variable—x2, x4 and x5. Hence the dimension of Nul A is 3. Also dim Col A 2 because A has two pivot columns. © 2012 Pearson Education, Inc. Slide 4.5- 23 © 2012 Pearson Education, Inc. Slide 4.4- 24 4 Vector Spaces 4.6 RANK © 2012 Pearson Education, Inc. THE ROW SPACE If A is an m n matrix, each row of A has n entries n and thus can be identified with a vector in . The set of all linear combinations of the row vectors is called the row space of A and is denoted by Row A. Each row has n entries, so Row A is a subspace of n . Since the rows of A are identified with the columns of AT, we could also write Col AT in place of Row A. © 2012 Pearson Education, Inc. Slide 4.6- 26 THE ROW SPACE Theorem 13: If two matrices A and B are row equivalent, then their row spaces are the same. If B is in echelon form, the nonzero rows of B form a basis for the row space of A as well as for that of B. © 2012 Pearson Education, Inc. Slide 4.6- 27 THE ROW SPACE Example 1: Find bases for the row space, the column space, and the null space of the matrix 8 2 5 1 3 5 3 11 19 1 7 13 0 17 1 5 7 1 5 3 Solution: To find bases for the row space and the column space, row reduce A to an echelon form: © 2012 Pearson Education, Inc. Slide 4.6- 28 THE ROW SPACE A 1 0 B 0 0 3 5 1 5 1 2 2 7 0 0 4 20 0 0 0 0 By Theorem 13, the first three rows of B form a basis for the row space of A (as well as for the row space of B). Thus Basis for Row A :{(1,3, 5,1,5),(0,1, 2,2, 7),(0,0,0, 4,20)} © 2012 Pearson Education, Inc. Slide 4.6- 29 THE ROW SPACE For the column space, observe from B that the pivots are in columns 1, 2, and 4. Hence columns 1, 2, and 4 of A (not B) form a basis for Col A: 2 5 0 1 3 1 Basis for Col A : , , 3 11 7 1 7 5 Notice that any echelon form of A provides (in its nonzero rows) a basis for Row A and also identifies the pivot columns of A for Col A. © 2012 Pearson Education, Inc. Slide 4.6- 30 THE ROW SPACE However, for Nul A, we need the reduced echelon form. Further row operations on B yield A 1 0 B C 0 0 © 2012 Pearson Education, Inc. 0 1 1 2 0 0 0 0 0 1 0 3 1 5 0 0 Slide 4.6- 31 THE ROW SPACE The equation Ax 0 is equivalent to Cx 0, that is, x1 x3 x5 0 x2 2 x3 3 x5 0 x4 5 x5 0 So x1 x3 x5 , x2 2 x3 3x5 , x4 5 x5, with x3 and x5 free variables. © 2012 Pearson Education, Inc. Slide 4.6- 32 THE ROW SPACE The calculations show that 1 1 2 3 Basis for Nul A : 1 , 0 0 5 0 1 Observe that, unlike the basis for Col A, the bases for Row A and Nul A have no simple connection with the entries in A itself. © 2012 Pearson Education, Inc. Slide 4.6- 33 THE RANK THEOREM Definition: The rank of A is the dimension of the column space of A. Since Row A is the same as Col AT, the dimension of the row space of A is the rank of AT. The dimension of the null space is sometimes called the nullity of A. Theorem 14: The dimensions of the column space and the row space of an m n matrix A are equal. This common dimension, the rank of A, also equals the number of pivot positions in A and satisfies the equation rank A dim Nul A n © 2012 Pearson Education, Inc. Slide 4.6- 34 THE RANK THEOREM number of number of number of pivot columns nonpivot columns columns © 2012 Pearson Education, Inc. Slide 4.6- 35 THE RANK THEOREM Example 2: a. If A is a 7 9 matrix with a two-dimensional null space, what is the rank of A? b. Could a 6 9 matrix have a two-dimensional null space? Solution: a. Since A has 9 columns, (rank A) 2 9 , and hence rank A 7. b. No. If a 6 9 matrix, call it B, has a twodimensional null space, it would have to have rank 7, by the Rank Theorem. © 2012 Pearson Education, Inc. Slide 4.6- 36 THE INVERTIBLE MATRIX THEOREM (CONTINUED) 6 But the columns of B are vectors in , and so the dimension of Col B cannot exceed 6; that is, rank B cannot exceed 6. Theorem: Let A be an n n matrix. Then the following statements are each equivalent to the statement that A is an invertible matrix. n m. The columns of A form a basis of . n A n. Col o. Dim Col A n p. rank A n © 2012 Pearson Education, Inc. Slide 4.6- 37 © 2012 Pearson Education, Inc. Slide 4.4- 38