d i E Dimension a l l u d Dr. Abdulla Eid b A College of - PowerPoint PPT Presentation

Section 4.5 d i E Dimension a l l u d Dr. Abdulla Eid b A College of Science . r D MATHS 211: Linear Algebra Dr. Abdulla Eid (University of Bahrain) Dimension 1 / 21

d i E Goal: a l 1 Define the dimension of a vector space. l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Dimension 2 / 21

Basis Definition 1 Let V be a vector space. Let B = { v 1 , v 2 , . . . , v n } be a basis . The number n is called the dimension of the vector space V , and denoted by d dim ( V ) . i E a l l u d b A . r D Note: This number n is independent of the chosen basis Theorem 2 All bases for a finite–dimensional vector space have the same number of vectors. Dr. Abdulla Eid (University of Bahrain) Dimension 3 / 21

d Lemma 3 i E Let V be a vector space. Let B = { v 1 , v 2 , . . . , v n } be a basis. a l l 1 If a set has more than n vectors, then it is linearly dependent. u d 2 If a set has fewer than n vectors, then it does not span V . b A . r D Dr. Abdulla Eid (University of Bahrain) Dimension 4 / 21

Standard Basis Note: The standard basis for R 2 is e 1 , e 2 , where e 1 = ( 1, 0 ) and e 2 = ( 0, 1 ) d So dim ( R 2 ) = 2. i E The standard basis for R 3 is e 1 , e 2 , e 3 , where a l e 1 = ( 1, 0, 0 ) , e 2 = ( 0, 1, 0 ) and e 3 = ( 0, 0, 1 ) l u d dim ( R 3 ) = 3. b The standard basis for R n is e 1 , e 2 , e 3 , . . . , e n , where A . e 1 = ( 1, 0, . . . , 0 ) , e 2 = ( 0, 1, . . . , 0 ) , e 3 = ( 0, 0, 1, 0, . . . , 0 ) and r D e n = ( 0, 0, . . . , 1 ) dim ( R n ) = n . The standard basis for P 2 is 1, X , X 2 . dim ( P 2 ) = 3. The standard basis for P n is 1, X , X 2 , . . . , X n . dim ( P n ) = n + 1. Dr. Abdulla Eid (University of Bahrain) Dimension 5 / 21

d i E The standard basis for Mat ( 2, 2, R ) is a l dim ( Mat ( 2, 2, R )) = 4. and in general dim ( Mat ( m , n , R )) = mn . l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Dimension 6 / 21

Example 4 Find bases for the subspace of R 3 given by the plane 2 x + 4 y − 3 z = 0 d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Dimension 7 / 21

Example 5 Find bases for the subspace of R 3 given by the plane x + z = 0 d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Dimension 8 / 21

Example 6 Find bases for the subspace of R 3 given by the plane x = 4 t , y = 2 t , z = − t . d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Dimension 9 / 21

Example 7 Find bases for the subspace of R 3 given by all vectors of the form ( a , b , c ) , where c = a − b . d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Dimension 10 / 21

Example 8 Find the dimension of the subspace W of R 4 , given by all vectors of the form ( 0, a , b , c ) . d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Dimension 11 / 21

Example 9 Find the dimension of the subspace W of R 4 , given by all vectors of the form ( a , b , c , d ) , where d = a + 2 b , c = 3 a − b . d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Dimension 12 / 21

Example 10 Find the dimension of the subspace W of R 4 , given by all vectors of the form ( a , b , c , d ) , where c = a , b = d = − a . d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Dimension 13 / 21

Example 11 Find the dimension of the subspace W of Mat ( n , n , R ) , given by all diagonal matrices. d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Dimension 14 / 21

Example 12 Find the dimension of the subspace W of Mat ( n , n , R ) , given by all symmetric matrices. d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Dimension 15 / 21

Example 13 Find the dimension of the subspace W of Mat ( n , n , R ) , given by all upper triangular matrices. d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Dimension 16 / 21

Example 14 Find the dimension of the subspace W of P n , given by all polynomials with a horizontal tangent at x = 0. d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Dimension 17 / 21

Infinite Dimensional Spaces Example 15 Let V = Maps ( R , R ) . Show that for every positive integer n , one can find n + 1 independent functions. d i E a l l u d b A . r D Conclusion: Maps ( R , R ) is infinite–dimensional. So do C i ( R , R ) and C ∞ ( R , R ) . Dr. Abdulla Eid (University of Bahrain) Dimension 18 / 21

Plus/Minus Theorem d Theorem 16 i E Let S be a nonempty set of vectors in a vector space V . a 1 If S is a linearly independent set, and v inV that is outside Span ( S ) , l l u then the set S ∪ { v } is still linearly independent. d b 2 If v is a vector in S that is expressible as a linear combination of other A vectors in S, then . r D Span ( S ) = Span ( S − { v } ) Dr. Abdulla Eid (University of Bahrain) Dimension 19 / 21

Example 17       1 2    ,  to produce a basis for R 3 . Enlarge the set 1 − 1    1 3 d  i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Dimension 20 / 21

Example 18       5 1    ,  to produce a basis for R 3 . Enlarge the set 3 − 1    0 2 d  i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Dimension 21 / 21