Linear Transformations: Standard Matrix

Transcription

1 Linear Transformations: Standard Matrix Linear Algebra Josh Engwer TTU November 5 Josh Engwer (TTU) Linear Transformations: Standard Matrix November 5 / 9

2 PART I PART I: STANDARD MATRIX (THE EASY CASE) COMPOSITION OF LINEAR TRANSFORMATIONS INVERSE OF A LINEAR TRANSFORMATION Josh Engwer (TTU) Linear Transformations: Standard Matrix November 5 / 9

3 Representing a Linear Transformation by a Matrix It s desirable to represent linear transformations as matrix-vector products: L(x) = Ax But how to systematically achieve this?? Consider linear transformation L : R R 3 such that ([ ]) x L(x, x ) = (x x, x + x, 3x ) L = x x x x + x 3x Josh Engwer (TTU) Linear Transformations: Standard Matrix November 5 3 / 9

4 Representing a Linear Transformation by a Matrix Consider linear transformation L : R R 3 such that ([ ]) x L(x, x ) = (x x, x + x, 3x ) L = x ([ ]) x L(x) = L x = x x x + x 3x = x = + x x x However, recall the standard basis for R : E = Then, L(e ) = Ae =, L(e ) = Ae = = {[ x x 3x 3 = A = ], x x x + x 3x [ [ x x ] = Ax ]} {e, e } L(e ) L(e ) Josh Engwer (TTU) Linear Transformations: Standard Matrix November 5 / 9

5 The Standard Matrix for a Linear Transformation Definition (Standard Matrix for a Linear Transformation) Let linear transformation L : R n R m s.t. L(x) = Ax x R n, where A R m n. Then A is called the standard matrix for linear transformation L. Proposition (Finding the Standard Matrix Easy Case) GIVEN: Linear Transformation L : R n R m s.t. L is explicitly defined. TASK: Find Standard Matrix A R m n s.t. L(x) = Ax () A = L(e ) L(e ) L(e n ) i.e. The columns of A are the images of the standard basis vectors for R n. Josh Engwer (TTU) Linear Transformations: Standard Matrix November 5 5 / 9

6 Composition of Linear Transformations Theorem (Composition of Linear Transformations) Let linear transformation L : R n R m s.t. L (x) = A x. Let linear transformation L : R m R p s.t. L (x) = A x. Then the composition L L is defined by (L L )(x) := L [L (x)]. Moreover, the composition L L is a linear transformation and PROOF: (L L )(x) = (A A )x (L L )(x + y) = L [L (x + y)] LT = L [L (x) + L (y)] LT = L [L (x)] + L [L (y)] = (L L )(x) + (L L )(y) (L L )(αx) = L [L (αx)] LT = L [αl (x)] LT = αl [L (x)] = α(l L )(x) L L is a linear transformation. (L L )(x) = L [L (x)] = L (A x) = A (A x) M = (A A )x QED Josh Engwer (TTU) Linear Transformations: Standard Matrix November 5 6 / 9

7 Inverse Linear Transformation Definition (Inverse Linear Transformation) Let linear transformation L : R n R n have identical domain & codomain. Then linear transformation L : R n R n is the inverse of L if (L L)(x) = x and (L L )(x) = x x R n L is called invertible if its inverse L exists. Theorem (Inverse Linear Transformation in terms of Standard Matrix) Let linear transformation L : R n R n s.t. L(x) = Ax. If L is invertible, then its inverse L : R n R n such that L (x) = A x Josh Engwer (TTU) Linear Transformations: Standard Matrix November 5 7 / 9

8 PART II PART II: STANDARD MATRIX (THE HARD CASE) STANDARD MATRIX FOR DERIVATIVE OF POLYNOMIALS STANDARD MATRIX FOR INTEGRAL OF POLYNOMIALS Josh Engwer (TTU) Linear Transformations: Standard Matrix November 5 8 / 9

9 Representing a Linear Transformation by a Matrix As seen in PART I, it s straightforward to find the standard matrix A for a linear transformation L when: (EASY CASE) L is defined explicitly in terms of components of x: ([ ]) x x x L = x x + x 3x - OR (EASY CASE) L is described in terms of images of standard basis vectors: ([ ]) L = ([ ]) 7, and L = 5 - BUT HOW TO HANDLE (HARD CASE) L is described in terms of images of non-std basis vectors: ([ ]) ([ ]) 5 L = 7, and L = Josh Engwer (TTU) Linear Transformations: Standard Matrix November 5 9 / 9

10 Standard Matrix The Hard Case (Motivation) ([ Given L = = = ( [ L () ]) = ( [ L () ([ ()L ([ ()L 7 ] [ + ( ) ] [ + () [ (, 7, ) T (5,, ) T ([ ]), and L = ]) ([ + ( )L ]) ([ + ()L ] ]) ]) = 7 5 = ]) = 7 ]) = 5 [ Gauss Jordan 5 (,, ) T (, 3, ) T, find A s.t. L(x) = Ax ( Superposition Principle ] ) Josh Engwer (TTU) Linear Transformations: Standard Matrix November 5 / 9

11 Standard Matrix The Hard Case (Motivation) ([ Given L = ]) = 7 [ ] (, 7, ) T (5,, ) T ([ = L(e ) = L A = 3 ]) =, and L ([ Gauss Jordan = L ([ x x ]) = ]) = 5, find A s.t. L(x) = Ax [ ] (,, ) T (, 3, ) T ([ ]), and L(e ) = L = 3 3 [ x x ] Josh Engwer (TTU) Linear Transformations: Standard Matrix November 5 / 9

12 Finding the Standard Matrix Hard Case Proposition (Finding the Standard Matrix Hard Case) GIVEN: Linear Transformation L s.t. images L(b ),..., L(b n ) are provided. TASK: () where B = {b,..., b n } is a non-standard basis for R n. Find Standard Matrix A R m n s.t. L(x) = Ax b L(b ) L(e ) Gauss Jordan I.. b n L(b n ). L(e n ) NOTE: Write non-standard basis vectors b,..., b n as row vectors. NOTE: Write the images L(b ),..., L(b n ) as transposes to avoid confusion. () A = L(e ) L(e n ) Josh Engwer (TTU) Linear Transformations: Standard Matrix November 5 / 9

13 Isomorphisms (Review) Vector spaces R, R, P 3 are all isomorphic to each other since there exists isomorphisms L, L as shown below: ([ ]) a Let L : R R a a s.t. L = a a a a Let L : P 3 R s.t. L ( a + a x + a x + a 3 x 3) = Isomorphic vector spaces are essentially the same but with representations by different mathematical objects (vectors, matrices, polynomials.) In particular, matrices & polynomials can be represented by vectors, which will be quite useful now... a a a a a 3 NOTATION: P 3 R means P 3 is isomorphic to R Josh Engwer (TTU) Linear Transformations: Standard Matrix November 5 3 / 9

14 Linear Transformations involving Calculus Differentiation & integration are in fact linear transformations: d dx [ f (x) + g(x)] dx = [ ] f (x) + g(x) = f (x) + g (x) f (x) dx + g(x) dx [ ] αf (x) = αf (x) d dx [ αf (x)] dx = α f (x) dx b a [ ] f (x) + g(x) dx = b a f (x) dx + b a g(x) dx b a [ ] αf (x) dx = α b a f (x) dx If one restricts attention to a vector space of polynomials, P n, this means that differentation & integration can be represented by standard matrices!! QUESTION: How to find the standard matrix for calculus of polynomials? ANSWER: Recognize that P n is isomorphic to R n+. Josh Engwer (TTU) Linear Transformations: Standard Matrix November 5 / 9

15 Linear Transformations involving Calculus WEX 6-3-: Find standard matrix A for linear transformation L : P P s.t. L(p) = p (t) Josh Engwer (TTU) Linear Transformations: Standard Matrix November 5 5 / 9

16 Linear Transformations involving Calculus WEX 6-3-: Find standard matrix A for linear transformation L : P P s.t. L(p) = p (t) d ] Find derivative of arbitrary polynomial in P : [a + bt + ct = b + ct dt a [ ] Apply isomorphisms: a + bt + ct b b and b + ct c c a [ ] [ ] a [ Then: A b b = = A is 3 = b b = c c c c [ ] e = A = = L(e ) L(e ) L(e 3 ) where e = t e 3 = t [ ] L(e ) =, L(e ) = d [ ] [ ] t =, L(e dt 3 ) = d ] [ ] [t = t dt ] Josh Engwer (TTU) Linear Transformations: Standard Matrix November 5 6 / 9

17 Finding the Standard Matrix for Calculus (Procedure) Proposition (Finding the Standard Matrix for Linear Transformations involving Calculus) GIVEN: Linear Transformation L : P n P m s.t. L involves calculus. TASK: Find Standard Matrix A s.t. L(x) = Ax () Compute images L(e ), L(e ),..., L(e n+ ). () Produce the isomorphisms of these images in P m to images in R m+ : [L(e )] E, [L(e )] E,..., [L(e n+ )] E (3) A = [L(e )] E [L(e )] E [L(e n+ )] E E = {, t, t,..., t n } {e, e,..., e n+ } is standard ordered basis for P n. E = {, t, t,..., t m } {e, e,..., e m+ } is standard ordered basis for P m. IMPORTANT: Always write polynomials in ascending order: 3 t + 5t Josh Engwer (TTU) Linear Transformations: Standard Matrix November 5 7 / 9

18 The Value of the Standard Matrix for Calculus By knowing the standard matrix for the derivative of a quadratic polynomial... d ] [ ] [a a [ ] + bt + ct b b = b + ct dt c c...derivatives of quadratics can be found simply with a matrix multiplication: d [ ] [ ] [ ] t t = t dt d [ ] [ 3t 7 dt d ] [ [ t dt d [ ] [ 5 dt ] ] ] [ ] 3 = 3 [ = [ ] = 8 ] 8t Josh Engwer (TTU) Linear Transformations: Standard Matrix November 5 8 / 9

19 Fin Fin. Josh Engwer (TTU) Linear Transformations: Standard Matrix November 5 9 / 9