Linear Algebra (wi1403lr) Lecture no.3

Transcription

1 Linear Algebra (wi1403lr) Lecture no.3 EWI / DIAM / Numerical Analysis group Matthias Möller 25/04/2014 M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

2 Review of lecture no Vector Equations vectors in R 2, R 3, and R n parallelogram rule = addition of two vectors in R 2 algebraic properties of vectors in R n linear combination of vectors in R n = addition of scaled vectors geometric interpretation of span{u} = line through the origin geometric interpretation of and span{u, v} = plane through the origin M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

3 Review of lecture no.2, cont d 1.4 The Matrix Equation Ax = b matrix product Ax = linear combination of the columns of matrix A using the corresponding entries in vector x as weights solution to Ax = b exists if and only if b is a linear combination of the columns of A computation of the matrix-vector product Ax properties of the matrix-vector product Ax M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

4 Learning objectives of lecture no.3 You will learn to describe solution sets of (non)homogeneous linear systems to interpret solution sets geometrically the concept of linear (in)dependence M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

5 Solution(s) to homogeneous systems Give a solution to the homogeneous linear system Ax = 0 with m n matrix A and zero vector in R m? M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

6 Nontrivial solution(s) to homogeneous systems Are there nontrivial solution(s) to the homogeneous linear system 2x 1 4x 2 = 0 3x 1 + 6x 2 = 0 M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

7 Nontrivial solution(s) to homogeneous systems Are there nontrivial solution(s) to the homogeneous linear system 2x 1 4x 2 = 0 3x 1 + 6x 2 = 0 Solution: x 2 is a free variable [ ] [ ] [ ] x1 2x2 2 x = = = x 2 = au, 1 a = x 2, u = x 2 x 2 [ ] 2 1 See EXAMPLE 1 for a 3 3 system. M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

8 Solution sets to homogeneous systems Take home lesson The homogeneous linear system Ax = 0 always has the trivial solution x = 0 has nontrivial solutions x 0 if and only if the equation has at least one free variable M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

9 Solution sets to homogeneous systems Exercise Find all nontrivial solutions to the homogeneous linear system x x x x 4 = x 5 0 x 6 M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

10 Solution sets to homogeneous systems Solution to Exercise Linear combination with free variables x 2, x 4 and x 6 as weights x = x x x Parametric vector form x = au + bv + cw, a, b, c R M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

11 Geometric interpretation of parametric form Quiz: Give the geometric interpretation 1 x = au, a R, u R 2 2 x = au, a R, u R 3 3 x = au, a R, u R n 4 x = au + bv, a, b R, u, v R 3, u v 5 x = au + bv + cw, a, b, c R, u, v, w R n, u v, u w, v w M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

12 Geometric interpretation of parametric form Quiz: Give the geometric interpretation 1 x = au, a R, u R 2 line in R 2 through 0 2 x = au, a R, u R 3 3 x = au, a R, u R n 4 x = au + bv, a, b R, u, v R 3, u v 5 x = au + bv + cw, a, b, c R, u, v, w R n, u v, u w, v w M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

13 Geometric interpretation of parametric form Quiz: Give the geometric interpretation 1 x = au, a R, u R 2 line in R 2 through 0 2 x = au, a R, u R 3 line in R 3 through 0 3 x = au, a R, u R n 4 x = au + bv, a, b R, u, v R 3, u v 5 x = au + bv + cw, a, b, c R, u, v, w R n, u v, u w, v w M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

14 Geometric interpretation of parametric form Quiz: Give the geometric interpretation 1 x = au, a R, u R 2 line in R 2 through 0 2 x = au, a R, u R 3 line in R 3 through 0 3 x = au, a R, u R n line in R n through 0 4 x = au + bv, a, b R, u, v R 3, u v 5 x = au + bv + cw, a, b, c R, u, v, w R n, u v, u w, v w M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

15 Geometric interpretation of parametric form Quiz: Give the geometric interpretation 1 x = au, a R, u R 2 line in R 2 through 0 2 x = au, a R, u R 3 line in R 3 through 0 3 x = au, a R, u R n line in R n through 0 4 x = au + bv, a, b R, u, v R 3, u v plane in R 3 through 0 we call this explicit description of the plane the parametric vector form in contrast to the implicit form αx 1 + βx 2 + γx 3 = 0 5 x = au + bv + cw, a, b, c R, u, v, w R n, u v, u w, v w M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

16 Geometric interpretation of parametric form Quiz: Give the geometric interpretation 1 x = au, a R, u R 2 line in R 2 through 0 2 x = au, a R, u R 3 line in R 3 through 0 3 x = au, a R, u R n line in R n through 0 4 x = au + bv, a, b R, u, v R 3, u v plane in R 3 through 0 we call this explicit description of the plane the parametric vector form in contrast to the implicit form αx 1 + βx 2 + γx 3 = 0 5 x = au + bv + cw, a, b, c R, u, v, w R n, u v, u w, v w object in R n through 0 M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

17 Solution(s) to inhomogeneous systems Are there solution(s) to the inhomogeneous linear system 2x 1 4x 2 = 2 3x 1 + 6x 2 = 3 M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

18 Solution(s) to inhomogeneous systems Are there solution(s) to the inhomogeneous linear system 2x 1 4x 2 = 2 3x 1 + 6x 2 = 3 Solution: x 2 is a free variable [ ] [ ] x x2 x = = = x 2 x 2 a = x 2, p = See EXAMPLE 3 for a 3 3 system. [ ] [ ] x 0 2 = p + au 1 [ ] 1, u = 0 [ ] 2 1 M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

19 Comparison of parametric vector forms Parametric vector form x = p + au, a R Homogeneous system Inhomogeneous system a = x 2, p = 0, u = a = x 2, p = [ ] 1, u = 0 [ ] 2 1 [ ] 2 1 Take home lesson: Solution to inhomogeneous system is x = particular solution + a homogeneous solution M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

20 Solutions to inhomogeneous systems Give the solutions to the inhomogeneous linear system 2x 1 + 2x 2 + 4x 3 = 8 4x 1 4x 2 8x 3 = 16 3x 2 3x 3 = 12 in parametric vector form. M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

21 Solutions to inhomogeneous systems Give the solutions to the inhomogeneous linear system 2x 1 + 2x 2 + 4x 3 = 8 4x 1 4x 2 8x 3 = 16 3x 2 3x 3 = 12 in parametric vector form. Solution: x 3 is a free variable 8 1 x = 4 + a }{{}}{{} p au M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

22 Uniqueness of solutions Quiz: A is a 3 2 matrix with two pivot positions. Ax = 0 has a nontrivial solution Yes No Ax = b has at least one solution for each b Yes No M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

23 Uniqueness of solutions Quiz: A is a 3 2 matrix with two pivot positions. Ax = 0 has a nontrivial solution Yes No two pivot positions each of the two columns is a pivot column no free variable no nontrivial solution Ax = b has at least one solution for each b Yes No M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

24 Uniqueness of solutions Quiz: A is a 3 2 matrix with two pivot positions. Ax = 0 has a nontrivial solution Yes No two pivot positions each of the two columns is a pivot column no free variable no nontrivial solution Ax = b has at least one solution for each b Yes No Theorem 4 (from 1.4): Ax = b has a solution for each b if and only if it has a pivot position in every row. A cannot have a pivot in every row (2 pivot positions, 3 rows) M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

25 Linear independence An indexed set of vectors {v 1,..., v p } in R n is termed linearly independent if the vector equation x 1 v 1 + x 2 v x p v p = 0 has only the trivial solution x 1 = x 2 = = x p = 0. An indexed set of vectors {v 1,..., v p } in R n is termed linearly dependent if there exist weights c 1,..., c p, not all zero, such that c 1 v 1 + c 2 v c p v p = 0. M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

26 Linear independence Theorem An indexed set S = {v 1,..., v p } of two or more vectors is linearly dependent if and only if at least one of the vectors in S is a linear combination of the others. In fact, if S is linearly dependent and v 1 0, then some v j (with j > 1) is a linear combination of the preceding vectors, v 1,..., v j 1. M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

27 Linear independence Theorem An indexed set S = {v 1,..., v p } of two or more vectors is linearly dependent if and only if at least one of the vectors in S is a linear combination of the others. In fact, if S is linearly dependent and v 1 0, then some v j (with j > 1) is a linear combination of the preceding vectors, v 1,..., v j 1. A word of caution: Not every vector in a linearly dependent set needs to be a linear combination of the preceding ones. M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

28 Linear independence Theorem An indexed set S = {v 1,..., v p } of two or more vectors is linearly dependent if and only if at least one of the vectors in S is a linear combination of the others. In fact, if S is linearly dependent and v 1 0, then some v j (with j > 1) is a linear combination of the preceding vectors, v 1,..., v j 1. A word of caution: Not every vector in a linearly dependent set needs to be a linear combination of the preceding ones. Homework: Read the proof of this theorem on page 60 of the book and try to understand it. If you still have question after having tried really hard, then ask me. M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

29 Linear independence Theorem If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set {v 1,..., v p } in R n is linearly dependent if p > n. Theorem If a set S = {v 1,..., v p } in R n contains the zero vector, then the set is linearly dependent. Reconsider your answers to exercises 15, 17, 19 M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18

30 Linear independence Quiz: True or false (then give a counterexample) If v 1,..., v 4 are in R 4 and v 3 = 2v 1 + v 2, then {v 1, v 2, v 3, v 4 } is linearly independent True False If v 1,..., v 4 are in R 4 and {v 1, v 2, v 3 } is linearly dependent, then {v 1, v 2, v 3, v 4 } is also linearly dependent True False If {v 1, v 2, v 3, v 4 } is a linearly independent set of vectors in R 4, then {v 1, v 2, v 3 } is also linearly independent True False M. Möller (EWI/NA group) LA (wi1403lr) 25/04/ / 18