Linear Transformations Marco Chiarandini Department of Mathematics - PowerPoint PPT Presentation

DM554 Linear and Integer Programming Lecture 8 Linear Transformations Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark

Linear Transformations Outline Coordinate Change 1. Linear Transformations 2. Coordinate Change 2

Linear Transformations Resume Coordinate Change • Linear dependence and independence • Determine linear dependency of a set of vertices, ie, find non-trivial lin. combination that equal zero • Basis • Find a basis for a linear space • Find a basis for the null space, range and row space of a matrix (from its reduced echelon form) • Dimension (finite, infinite) • Rank-nullity theorem 3

Linear Transformations Outline Coordinate Change 1. Linear Transformations 2. Coordinate Change 4

Linear Transformations Linear Transformations Coordinate Change Definition (Linear Transformation) Let V and W be two vector spaces. A function T : V → W is linear if for all u , v ∈ V and all α ∈ R : 1. T ( u + v ) = T ( u ) + T ( v ) 2. T ( α u ) = α T ( u ) A linear transformation is a linear function between two vector spaces • If V = W also known as linear operator • Equivalent condition: T ( α u + β v ) = α T ( u ) + β T ( v ) • for all 0 ∈ V , T ( 0 ) = 0 5

Linear Transformations Coordinate Change Example (Linear Transformations) • vector space V = R , F 1 ( x ) = px for any p ∈ R ∀ x , y ∈ R , α, β ∈ R : F 1 ( α x + β y ) = p ( α x + β y ) = α ( px ) + β ( px ) = α F 1 ( x ) + β F 1 ( y ) • vector space V = R , F 1 ( x ) = px + q for any p , q ∈ R or F 3 ( x ) = x 2 are not linear transformations T ( x + y ) � = T ( x ) + T ( y ) ∀ x , y ∈ R • vector spaces V = R n , W = R m , m × n matrix A , T ( x ) = A x for x ∈ R n T ( u + v ) = A ( u + v ) = A u + A v = T ( u ) + T ( v ) T ( α u ) = A ( α u ) = α A u = α T ( u ) 6

Linear Transformations Coordinate Change Example (Linear Transformations) • vector spaces V = R n , W : f : R → R . T : R n → W :     u 1 u 2     T ( u ) = T  = p u 1 , u 2 ,..., u n = p u   .   .     .    u n p u 1 , u 2 ,..., u n = u 1 x 1 + u 2 x 2 + u 3 x 3 + · · · + u n x n p u + v ( x ) = · · · = ( p u + p v )( x ) p α u ( x ) = · · · = α p u ( x ) 7

Linear Transformations Linear Transformations and Matrices Coordinate Change • any m × n matrix A defines a linear transformation T : R n → R m � T A • for every linear transformation T : R n → R m there is a matrix A such that T ( v ) = A v � A T Theorem Let T : R n → R m be a linear transformation and { e 1 , e 2 , . . . , e n } denote the standard basis of R n and let A be the matrix whose columns are the vectors T ( e 1 ) , T ( e 2 ) , . . . , T ( e n ) : that is, � � A = T ( e 1 ) T ( e 2 ) . . . T ( e n Then, for every x ∈ R n , T ( x ) = A x . Proof: write any vector x ∈ R n as lin. comb. of standard basis and then make the image of it. 8

Example T : R 3 → R 3       x x + y + z  = T y x − y      z x + 2 y − 3 z • The image of u = [ 1 , 2 , 3 ] T can be found by substitution: T ( u ) = [ 6 , − 1 , − 4 ] T . • to find A T :       1 1 1 T ( e 1 ) = 1 T ( e 2 ) = − 1 T ( e 3 ) = 0       1 2 − 3   1 1 1 A = [ T ( e 1 ) T ( e 2 ) T ( e n )] = 1 − 1 0   1 2 − 3 T ( u ) = A u = [ 6 , − 1 , − 4 ] T .

Linear Transformations Linear Transformation in R 2 Coordinate Change • We can visualize them! • Reflection in the x axis: � x � x � � � 1 � 0 T : �→ A T = y − y 0 − 1 • Stretching the plane away from the origin � � � � 2 0 x T ( x ) = 0 3 y 10

R • Rotation anticlockwise by an angle θ y 1 e 2 T ( e 2 ) θ T ( e 1 ) θ x e 1 ( 0 , 0 ) 1 we search the images of the standard basis vector e 1 , e 2 � a � � d � T ( e 1 ) = , T ( e 1 ) = c b they will be orthogonal and with length 1. � a b � � cos θ − sin θ � A = = c d sin θ cos θ For π/ 4: � 1 2 − 1 � � � � � a b cos θ − sin θ √ √ 2 A = = = 1 1 c d sin θ cos θ √ √ 2 2 (the matrix A is correct, in the lecture, I made a mistake placing the θ angle on the other side of e 2 )

Linear Transformations Identity and Zero Linear Transformations Coordinate Change • For T : V → V the linear transformation such that T ( v ) = v is called the identity. • if V = R n , the matrix A T = I (of size n × n ) • For T : V → W the linear transformation such that T ( v ) = 0 is called the zero transformation. • If V = R n and W = R m , the matrix A T is an m × n matrix of zeros. 12

Linear Transformations Composition of Linear Transformations Coordinate Change • Let T : V → W and S : W → U be linear transformations. The composition of ST is again a linear transformation given by: ST ( v ) = S ( T ( v )) = S ( w ) = u where w = T ( v ) T S • ST means do T and then do S : V − → W − → U • if T : R n → R m and S : R m → R p in terms of matrices: ST ( v ) = S ( T ( v )) = S ( A T v ) = A S A T v note that composition is not commutative 13

Linear Transformations Combinations of Linear Transformations Coordinate Change • If S , T : V → W are linear transformations between the same vector spaces, then S + T and α S , α ∈ R are linear transformations. • hence also α S + β T , α, β ∈ R is 14

Linear Transformations Inverse Linear Transformations Coordinate Change • If V and W are finite-dimensional vector spaces of the same dimension, then the inverse of a lin. transf. T : V → W is the lin. transf such that T − 1 ( T ( v )) = v • In R n if T − 1 exists, then its matrix satisfies: T − 1 ( T ( v )) = A T − 1 A T v = I v that is, T − 1 exists iff ( A T ) − 1 exists and A T − 1 = ( A T ) − 1 (recall that if BA = I then B = A − 1 ) • In R 2 for rotations: � � � � cos ( − θ ) − sin ( − θ ) cos θ sin θ A T − 1 = = sin ( − θ ) cos ( − θ ) − sin θ cos θ 15

Linear Transformations Coordinate Change Example Is there an inverse to T : R 3 → R 3       x x + y + z  = T y x − y      z x + 2 y − 3 z   1 1 1 A = 1 − 1 0   1 2 − 3 Since det ( A ) = 9 then the matrix is invertible, and T − 1 is given by the matrix: 3 u + 5 1 9 v + 1         3 5 1 u 9 w A − 1 = 1 T − 1  = 3 u − 4 1 9 v + 1 3 − 4 1 v 9 w        9 1 3 u + 1 9 v − 2 3 − 1 − 2 w 9 w 16

Linear Transformations Linear Transformations from V to W Coordinate Change Theorem Let V be a finite-dimensional vector space and let T be a linear transformation from V to a vector space W . Then T is completely determined by what it does to a basis of V . Proof (unique representation in V implies unique representation in T ) 17

Linear Transformations Coordinate Change • If both V and W are finite dimensional vector spaces, then we can find a matrix that represents the linear transformation: • suppose V has dim ( V ) = n and basis B = { v 1 , v 2 , . . . , v n } and W has dim ( W ) = m and basis S = { w 1 , w 2 , . . . , w m } ; • coordinates of v ∈ V are [ v ] B coordinates of T ( v ) ∈ W are [ T ( v )] S • we search for a matrix A such that: [ T ( v )] S = A [ v ] B • we find it by: [ T ( v )] S = a 1 [ T ( v 1 )] S + a 2 [ T ( v 2 )] S + · · · + a n [ T ( v n )] S = [[ T ( v 1 )] S [ T ( v 2 )] S · · · [ T ( v n )] S ] [ v ] B where [ v ] B = [ a 1 , a 2 , . . . , a n ] T 18

Linear Transformations Range and Null Space Coordinate Change Definition (Range and null space) T : V → W . The range R ( T ) of T is: R ( T ) = { T ( v ) | v ∈ V } and the null space (or kernel) N ( T ) of T is N ( T ) = { v ∈ V | T ( v ) = 0 } • the range is a subspace of W and the null space of V . • Matrix case, T : R n → R m R ( T ) = R ( A ) N ( T ) = N ( A ) • Rank-nullity theorem: rank ( T ) = dim ( R ( T )) nullity ( T ) = dim ( N ( T )) rank ( T ) + nullity ( T ) = dim ( V ) 19

Linear Transformations Coordinate Change Example Construct a linear transformation T : R 3 → R 3 with     1    : t ∈ R N ( T ) =  t 2  , R ( T ) = xy -plane.  3 20

Linear Transformations Outline Coordinate Change 1. Linear Transformations 2. Coordinate Change 21

Linear Transformations Coordinates Coordinate Change Recall: Definition (Coordinates) If S = { v 1 , v 2 , . . . , v n } is a basis of a vector space V , then • any vector v ∈ V can be expressed uniquely as v = α 1 v 1 + · · · + α n v n • and the real numbers α 1 , α 2 , . . . , α n are the coordinates of v wrt the basis S . To denote the coordinate vector of v in the basis S we use the notation   α 1 α 2   [ v ] S =  .  .   .   α n S • In the standard basis the coordinates of v are precisely the components of the vector v : v = v 1 e 1 + v 2 e 2 + · · · + v n e n • How to find coordinates of a vector v wrt another basis? 22