Vector Spaces Linear Independence, Bases and Dimension Marco - PowerPoint PPT Presentation

DM559 Linear and Integer Programming Lecture 7 Vector Spaces Linear Independence, Bases and Dimension Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark

Vector Spaces and Subspaces Linear independence Outline Bases and Dimension 1. Vector Spaces and Subspaces 2. Linear independence 3. Bases and Dimension 5

Vector Spaces and Subspaces Linear independence Outline Bases and Dimension 1. Vector Spaces and Subspaces 2. Linear independence 3. Bases and Dimension 6

Vector Spaces and Subspaces Linear independence Premise Bases and Dimension • We move to a higher level of abstraction • A vector space is a set with an addition and scalar multiplication that behave appropriately, that is, like R n • Imagine a vector space as a class of a generic type (template) in object oriented programming, equipped with two operations. 7

Vector Spaces and Subspaces Linear independence Vector Spaces Bases and Dimension Definition (Vector Space) A (real) vector space V is a non-empty set equipped with an addition and a scalar multiplication operation such that for all α, β ∈ R and all u , v , w ∈ V : 1. u + v ∈ V (closure under addition) 2. u + v = v + u (commutative law for addition) 3. u + ( v + w ) = ( u + v ) + w (associative law for addition) 4. there is a single member 0 of V , called the zero vector, such that for all v ∈ V , v + 0 = v 5. for every v ∈ V there is an element w ∈ V , written − v , called the negative of v , such that v + w = 0 6. α v ∈ V (closure under scalar multiplication) 7. α ( u + v ) = α u + α v (distributive law) 8. ( α + β ) v = α v + β v (distributive law) 9. α ( β v ) = ( αβ ) v (associative law for vector multiplication) 10. 1 v = v 8

Vector Spaces and Subspaces Linear independence Examples Bases and Dimension • set R n • but the set of objects for which the vector space defined is valid are more than the vectors in R n . • set of all functions F : R → R . We can define an addition f + g : ( f + g )( x ) = f ( x ) + g ( x ) and a scalar multiplication α f : ( α f )( x ) = α f ( x ) • Example: x + x 2 and 2 x . They can represent the result of the two operations. • What is − f ? and the zero vector? 9

Vector Spaces and Subspaces Linear independence Bases and Dimension The axioms given are minimum number needed. Other properties can be derived: For example: ( − 1 ) x = − x Proof: 0 = 0 x = ( 1 + ( − 1 )) x = 1 x + ( − 1 ) x = x + ( − 1 ) x Adding − x on both sides: − x = − x + 0 = − x + x + ( − 1 ) x = ( − 1 ) x which proves that − x = ( − 1 ) x . Try the same with − f . 10

Vector Spaces and Subspaces Linear independence Examples Bases and Dimension • V = { 0 } • the set of all m × n matrices • the set of all infinite sequences of real numbers, y = { y 1 , y 2 , . . . , y n , . . . , } , y i ∈ R . ( y = { y n } , n ≥ 1) – addition of y = { y 1 , y 2 , . . . , y n , . . . , } and z = { z 1 , z 2 , . . . , z n , . . . , } then: y + z = { y 1 + z 1 , y 2 + z 2 , . . . , y n + z n , . . . , } – multiplication by a scalar α ∈ R : α y = { α y 1 , α y 2 , . . . , α y n , . . . , } • set of all vectors in R 3 with the third entry equal to 0 (verify closure):    �  x �   � W = y x , y ∈ R   � � 0   � 11

Vector Spaces and Subspaces Linear independence Linear Combinations Bases and Dimension Definition (Linear Combination) For vectors v 1 , v 2 , . . . , v k in a vector space V , the vector v = α 1 v 1 + α 2 v 2 + . . . + α k v k is called a linear combination of the vectors v 1 , v 2 , . . . , v k . The scalars α i are called coefficients. • To find the coefficients that given a set of vertices express by linear combination a given vector, we solve a system of linear equations. • If F is the vector space of functions from R to R then the function f : x �→ 2 x 2 + 3 x + 4 can be expressed as a linear combination of: g : x �→ x 2 , h : x �→ x , k : x �→ 1 that is: f = 2 g + 3 h + 4 k • Given two vectors v 1 and v 2 , is it possible to represent any point in the Cartesian plane? 12

Vector Spaces and Subspaces Linear independence Subspaces Bases and Dimension Definition (Subspace) A subspace W of a vector space V is a non-empty subset of V that is itself a vector space under the same operations of addition and scalar multiplication as V . Theorem Let V be a vector space. Then a non-empty subset W of V is a subspace if and only if both the following hold: • for all u , v ∈ W , u + v ∈ W ( W is closed under addition) • for all v ∈ W and α ∈ R , α v ∈ W ( W is closed under scalar multiplication) ie, all other axioms can be derived to hold true 13

Vector Spaces and Subspaces Linear independence Bases and Dimension Example • The set of all vectors in R 3 with the third entry equal to 0. • The set { 0 } is not empty, it is a subspace since 0 + 0 = 0 and α 0 = 0 for any α ∈ R . Example In R 2 , the lines y = 2 x and y = 2 x + 1 can be defined as the sets of vectors: �� � � � �� � � � x x � � S = � y = 2 x , x ∈ R U = � y = 2 x + 1 , x ∈ R � � y y S = { x | x = t v , t ∈ R } U = { x | x = p + t v , t ∈ R } � � � � 1 0 v = , p = 2 1 14

Vector Spaces and Subspaces Linear independence Bases and Dimension Example (cntd) 1. The set S is non-empty, since 0 = 0 v ∈ S . 2. closure under addition: � � � � 1 1 u = s ∈ S , w = t ∈ S , for some s , t ∈ R 2 2 u + w = s v + t v = ( s + t ) v ∈ S since s + t ∈ R 3. closure under scalar multiplication: � 1 � u = s ∈ S for some s ∈ R , α ∈ R 2 α u = α ( s ( v )) = ( α s ) v ∈ S since α s ∈ R Note that: • u , w and α ∈ R must be arbitrary 15

Vector Spaces and Subspaces Linear independence Bases and Dimension Example (cntd) 1. 0 �∈ U 2. U is not closed under addition: � 0 � � 1 � � 0 � � 1 � � 1 � ∈ U , ∈ U but + = �∈ U 1 3 1 3 4 3. U is not closed under scalar multiplication � 0 � � 0 � � 0 � ∈ U , 2 ∈ R but 2 = �∈ U 1 1 2 Note that: • proving just one of the above couterexamples is enough to show that U is not a subspace • it is sufficient to make them fail for particular choices • a good place to start is checking whether 0 ∈ S . If not then S is not a subspace 16

Vector Spaces and Subspaces Linear independence Bases and Dimension Theorem A non-empty subset W of a vector space is a subspace if and only if for all u , v ∈ W and all α, β ∈ R , we have α u + β v ∈ W . That is, W is closed under linear combination. 17

Vector Spaces and Subspaces Linear independence Bases and Dimension Geometric interpretation: y y u w w u x x ( 0 , 0 ) ( 0 , 0 ) � The line y = 2 x + 1 is an affine subset, a „translation“ of a subspace 18

Vector Spaces and Subspaces Linear independence Null space of a Matrix is a Subspace Bases and Dimension Theorem For any m × n matrix A , N ( A ) , ie, the solutions of A x = 0 , is a subspace of R n Proof 1. A 0 = 0 = ⇒ 0 ∈ N ( A ) 2. Suppose u , v ∈ N ( A ) , then u + v ∈ N ( A ) : A ( u + v ) = A u + A v = 0 + 0 = 0 3. Suppose u ∈ N ( A ) and α ∈ R , then α u ∈ N ( A ) : A ( α u ) = A ( α u ) = α A u = α 0 = 0 The set of solutions S to a general system A x = b is not a subspace of R n because 0 �∈ S 19

Vector Spaces and Subspaces Linear independence Affine subsets Bases and Dimension Definition (Affine subset) If W is a subspace of a vector space V and x ∈ V , then the set x + W defined by x + W = { x + w | w ∈ W } is said to be an affine subset of V . The set of solutions S to a general system A x = b is an affine subspace , indeed recall that if x 0 is any solution of the system S = { x 0 + z | z ∈ N ( A ) } 20

Vector Spaces and Subspaces Linear independence Range of a Matrix is a Subspace Bases and Dimension Theorem For any m × n matrix A , R ( A ) = { A x | x ∈ R n } is a subspace of R m Proof 1. A 0 = 0 = ⇒ 0 ∈ R ( A ) 2. Suppose u , v ∈ R ( A ) , then u + v ∈ R ( A ) : ... 3. Suppose u ∈ R ( A ) and α ∈ R , then α u ∈ R ( A ) : ... 21

Vector Spaces and Subspaces Linear independence Linear Span Bases and Dimension • If v = α 1 v 1 + α 2 v 2 + . . . + α k v k and w = β 1 v 1 + β 2 v 2 + . . . + β k v k , then v + w and s v , s ∈ R are also linear combinations of the vectors v 1 , v 2 , . . . , v k . • The set of all linear combinations of a given set of vectors of a vector space V forms a subspace : Definition (Linear span) Let V be a vector space and v 1 , v 2 , . . . , v k ∈ V . The linear span of X = { v 1 , v 2 , . . . , v k } is the set of all linear combinations of the vectors v 1 , v 2 , . . . , v k , denoted by Lin ( X ) , that is: Lin ( { v 1 , v 2 , . . . , v k } ) = { α 1 v 1 + α 2 v 2 + . . . + α k v k | α 1 , α 2 , . . . , α k ∈ R } Theorem If X = { v 1 , v 2 , . . . , v k } is a set of vectors of a vector space V , then Lin ( X ) is a subspace of V and is also called the subspace spanned by X . It is the smallest subspace containing the vectors v 1 , v 2 , . . . , v k . 22