Linear algebra. NEU 466M Instructor: Professor Ila R. Fiete Spring 2016

Transcription

1 Linear algebra NEU M Instructor: Professor Ila R. Fiete Spring 01

2 NotaBon Matrices: upper-case A, B, U, W Vector: bold, (usually) lower-case x, y, v, w x! x (handwribng: ) Elements of matrix, vector: lower-case a ij,b i,v j,u kl Scalar numbers: lower-case, no indices a, b, c,,

3 Vectors and matrices v = v 1 v. v m size (m x 1) column vector v i R v R m A = a 11 a 1 a 1m a 1 a a m a n1 a n a nm size (n x m) matrix A R n m

4 What is a vector? geometric view v = v 1 v. v m v v 1 v size (m x 1) column vector plotv in matlab

5 Vector length v = v 1 v. v m Length (norm): v = q v 1 + v + v m v

6 Vector-scalar product v = v 1 v. v m geometric view v v same direcbon, different length

7 Sum of vectors v, u R m v + u = v 1 + u 1 v + u. v m + u m geometric view u v v + u v u u v Adding vectors: stacking them end-to-end

8 Unit vector: any vector of length 1 v = q v 1 + v + v m == v ux t m vi =1 i=1 ê ê ê 1 Every point on (m-1)-dimensional sphere of unit radius in m-dim space is a unit vector

9 Vector, matrix transpose v = v 1 v. v m v T =[v 1 v v m ] size (m x 1) column vector size (1 x m) row vector A = a 11 a 1 a 1m a 1 a a m a n1 a n a nm A T = a 11 a n1 a 1 a m a 1m a nm size (n x m) matrix size (m x n) matrix

10 Vector norm as an inner product v T v =[v 1 v v m ] v 1 v. v m = mx i=1 v i = v

11 Inner product (dot product) v, u R m u T v =[u 1 u u m ] v 1 v. v m = X i u i v i Geometric view: projecbon of v on u, Bmes norm of u: u T v = u v cos( ) v u u v cos( )

12 Example: Inner product (dot product) u = apple 1 0 u, v R unit vector along x-axis, v = apple v1 v u T v = v 1 v u 1

13 Inner product (dot product) v, u R m Example: u? v u T v = u v cos( ) =0 v u

14 System of equabons n equabons in m unknowns (v 1, v m ): a 11 v a 1m v m = b 1 a 1 v a m v m = b a n1 v a nm v m = b n

15 System of equabons n equabons in m unknowns (v 1, v m ): a 11 v a 1m v m = b 1 a 1 v a m v m = b a n1 v a nm v m = b n a 11 a 1 a 1m a 1 a a m a n1 a n a nm v 1 v. v m (n x m) (m x 1) (n x 1) = Av = b b 1 b.. b n

16 System of equabons: when does unique solubon exist? n equabons in m unknowns: generically, a unique solubon exists when same number of constraints (n) as unknowns (m): Thus, n=m or A is square. a 11 a 1m a 1 a m a m1 a mm v 1 v. v m b m (m x m) (m x 1) (n x 1) = Av = b b 1 b.. (m x m) (m x 1) (m x 1) m m = this is an algebraic view. Bme for some geometric insight.

17 Geometric view: when does a unique solubon exist? Start with -dimensional problem: unknowns, equabons equabon of a line a 11 x 1 + a 1 x = b 1 a 1 x 1 + a x = b unknowns x 1,x x a 1 x 1 + a x = b a 11 x 1 + a 1 x = b 1 solubon: at intersecbon where both equabons hold x 1

18 Geometric view: when does a unique solubon exist? Start with -dimensional problem: unknowns, equabons equabon of a line a 11 x 1 + a 1 x = b 1 a 1 x 1 + a x = b unknowns x 1,x x a 1 x 1 + a x = b a 11 x 1 + a 1 x = b 1 solubon: at intersecbon where both equabons hold Generically two infinite lines in D space intersect at a (single) locabon thus (unique) solubon exists. x 1

19 Geometric view: when does a unique solubon not exist? 1. Offset parallel lines: no solubon exists x a 1 x 1 + a x = b a 11 x 1 + a 1 x = b 1 b /a 1 b 1 /a 11 x 1

20 Algebra: when does a unique solubon not exist? 1. Offset parallel lines: no solubon exists x a 1 x 1 + a x = b a 11 x 1 + a 1 x = b 1 b /a 1 b 1 /a 11 x 1 a 1 /a = a 11 /a 1 a 11 a = a 1 a 1 equal slopes a 11 a a 1 a 1 =0

21 Algebra: when does a unique solubon not exist?. Aligned parallel lines: infinitely many solubons x a 1 x 1 + a x = b a 11 x 1 + a 1 x = b 1 b 1 /a 11 b /a 1 x 1 a 11 a a 1 a 1 =0 b 1 /a 11 = b /a 1 equal slopes equal intercepts

22 Algebraic view: existence of unique solubon in terms of coefficient matrix A A = apple a11 a 1 a 1 a determinant: det(a) a 11 a a 1 a 1 -dim system of equabons with square coefficient matrix A has a unique solubon when: det(a) = 0 Same condibon for m-dim system of equabons with square coefficient matrix.

23 Linear system: possibilibes 1 unique solubon No solubons Infinitely many solubons