Mathematical Principles for Scientific Computing and Visualization

Transcription

1 Mathematical Principles for Scientific Computing and Visualization Chapter 4: Background Numerical Linear Algebra Gerald Farin & Dianne Hansford CRC Press, Taylor & Francis Group, An A K Peters Book c 2008 Farin & Hansford Math Principles of SCV 1 / 30

2 Outline 1 Introduction 2 Linear Spaces and Vectors 3 Linear Independence and Subspaces 4 Linear Maps and Matrices 5 Lengths and Volumes 6 Matrix Fun Farin & Hansford Math Principles of SCV 2 / 30

3 Matrices! Matrix rectangular array of numbers Digital images Google matrix Transformation matrices Important tool in scientific computing Farin & Hansford Math Principles of SCV 3 / 30

4 Vectors u+2v u u 0.5u+0.5v v v Two-dimensional (2D) vectors live in R 2 Notation: u = [ u1 u 2 ] Linear combination w = su+tv Farin & Hansford Math Principles of SCV 4 / 30

5 Linear Space Set in which any two elements can be linearly combined and the result is again in the space digital images of the same size differentiable functions quadrilaterals in the plane 2D vectors Not a linear space: set of 2D vectors u with u 1 >= 0 Why? Farin & Hansford Math Principles of SCV 5 / 30

6 Linearly Dependent Vectors u = 1 v = 1 w = w = 2u v Are e 1,e 2,e 3 linearly dependent? Farin & Hansford Math Principles of SCV 6 / 30

7 Dimension and Basis n linearly independent vectors in vector space R n, u = u 1 u 2 u 3. u n , , ,, Dimension of a linear space: largest number of linearly independent vectors in it Basis: n linearly independent vectors in a linear space of dimension n This means that... For every v in R n there exists unique α i such that v = α 1 u 1 +α 2 u α n u n Farin & Hansford Math Principles of SCV 7 / 30

8 Basis Example 0 1 u = 1 and v = Do these vectors form a basis for R 3? Farin & Hansford Math Principles of SCV 8 / 30

9 Linear Space Linearly independent vectors u 1,u 2,...,u m in dimension n linear space S form linear space U of dimension m U is subspace of S U is spanned by the u i All u = s 1 u s m u m Example: 0 1 u 1 = 1 and u 2 = Farin & Hansford Math Principles of SCV 9 / 30

10 Linear Map Transform u to u [ u 1 ] [ ] 2u1 u 2 = 0.5u 2 In matrix form: u = Au = [ ][ u1 u 2 ] [ ] 2u1 = 0.5u 2 u called image vector Farin & Hansford Math Principles of SCV 10 / 30

11 Types of Linear Maps SCIVIZ SCIVIZ Shear A u = [ ][ u1 u 2 ] [ ] u1 +u = 2 u 2 Can you name some other linear maps? Farin & Hansford Math Principles of SCV 11 / 30

12 Linear Maps in Higher Dimensions Linear map v = Au m n matrix A m rows and n columns u R n mapped to v R m ith element of v formed as dot product v i = a i,1 u 1 +a i,2 u a i,n u n i = 1,...,m [ ] = [ ] 1 1 Farin & Hansford Math Principles of SCV 12 / 30

13 Linearity Property u Au 0.5u+0.5v v 0.5Au+0.5Av Av Linear maps preserve linear relationships A(su+tv) = sau+tav Nonlinear maps do not preserve linear relationships Try this example : [ u1 u 2 ] [ u1 u 2 2 ] with u = [ ] 1,v = 1 [ ] 0,s = 2,t = 3 2 Farin & Hansford Math Principles of SCV 13 / 30

14 Another Linearity Property [A+B]u = Au+Bu Identity matrix: I = Iu = u Farin & Hansford Math Principles of SCV 14 / 30

15 Combining Linear Maps v = Au and w = Bv j w = B A u n A Matrix product C = BA n B i C [ ] = [ ] Farin & Hansford Math Principles of SCV 15 / 30

16 Matrix Product II A different way to do matrix multiplication BA = b 1 a T b na T n b r rth column of B a T r the rth row of A [ ] BA = [ ] [ ] [ ] 1 [1 ] 0 [2 ] 1 [ 1 ] = [ ] [ ] = [ ] = [ ] Dyadic matrix b i a T i Farin & Hansford Math Principles of SCV 16 / 30

17 Order Matters AB BA Dimensions! Example: Rotations in 3D A 90 rotation about e 1 B 90 rotation about e 2 Sketch ABe 1 = e 2 BAe 1 = e 3 Farin & Hansford Math Principles of SCV 17 / 30

18 Rank Square matrix A mapping R n to R n n linearly independent vectors u 1,...,u n Mapped to image vectors Au 1,...,Au n Image vectors span a subspace of R n Rank: dimension of this subspace Also: number of linearly independent vectors in A Full rank: subspace is again all of R n Farin & Hansford Math Principles of SCV 18 / 30

19 Inverse Given: full rank square matrix A Find: matrix B such that BAu = u Inverse matrix A 1 undoes action BA = I A 1 A = I Example: Rotating by α degrees [ ] cosα sinα A = sinα cosα Then rotate by α degrees: (AB) 1 = B 1 A 1 B = [ ] cosα sinα sinα cosα Farin & Hansford Math Principles of SCV 19 / 30

20 Singular Matrices Non-full-rank matrices = = Farin & Hansford Math Principles of SCV 20 / 30

21 Dot Product Also called scalar product u v = u 1 v u n v n = u T u Farin & Hansford Math Principles of SCV 21 / 30

22 Vector Length u = u u2 n = u T u unit vector u = 1 All unit vectors in R n form what n-dimensional object? Farin & Hansford Math Principles of SCV 22 / 30

23 Angle between Vectors cos(u,v) = u v u v cos(θ) θ -1 Two perpendicular vectors enclose 90 o angle u v = 0 Perpendicular vectors more stable computations Farin & Hansford Math Principles of SCV 23 / 30

24 Orthogonal Vectors Unit vectors u 1,...,u n in R n Mutually orthogonal or orthonormal { 1 if i = j u i u j = 0 if i j δ i,j Kronecker delta Farin & Hansford Math Principles of SCV 24 / 30

25 Transpose Matrix Matrix A with elements a i,j Transpose matrix A T with elements a i,j row i of A T is column i of A a i,j = a j,i A TT = A Farin & Hansford Math Principles of SCV 25 / 30

26 Orthogonal Matrix Matrix U with orthonormal column vectors U T U = I Recall definition of the inverse matrix: U 1 U = I U 1 = U T What is the most obvious example of an orthogonal matrix? Another example? (Hint: a simple linear map) Farin & Hansford Math Principles of SCV 26 / 30

27 Diagonal Matrices d d 2 D = d n 1/d D 1 0 1/d 2 = /d n DD 1 = I Farin & Hansford Math Principles of SCV 27 / 30

28 Volume I What is the volume formed by [ ] 1 0 and [ ] 0? 1 Generalize: volume of n dimensional column vectors of I is 1 What is the volume formed by [ ] r 0 and [ ] 0? s Volume of n dimensional vectors in diagonal matrix D D = d 1 d 2... d n Determinant Also det(d) Farin & Hansford Math Principles of SCV 28 / 30

29 Volume II Orthogonal matrix U Rotate column vectors to I by a rotation Rotations do not change volumes U = 1 Singular matrix: A = 0 What is an example of a singular diagonal matrix? More of determinants in Solving Linear Systems and Eigen-Problems Farin & Hansford Math Principles of SCV 29 / 30

30 Matrix Fun Commutative Laws Distributive Laws Rules for Transpose Matrices Rules for Inverse Matrices A+B = A+B AB = BA A(B +C) =?? (B +C)A =?? (A+B) T = A T +B T (AB) T =?? A 1T = A T 1 (AB) 1 =?? (A+B) 1 A 1 +B 1 Farin & Hansford Math Principles of SCV 30 / 30