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Chapter 4 - Part 2

Linear Algebra. Chapter 4 - Part 2. Vector Spaces. 4.2 Linear Combinations of Vectors. Definition Let v 1 , v 2 , …, v m be vectors in a vector space V . We say that , is a ……………………… of ……………… , if there exist ……………..… such that v can be write as ………………………..….

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Chapter 4 - Part 2

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  1. Linear Algebra Chapter 4 - Part 2 Vector Spaces

  2. 4.2 Linear Combinations of Vectors Definition Let v1, v2, …, vm be vectors in a vector space V. We say that , is a ……………………… of ……………… , if there exist ……………..… such that v can be write as ………………………..… Example 1 The vector (7, 3, 2) is a linear combination of the vectors (1, 3, 0), (2, -3, 1) since: Solution Ch04_2

  3. Example 2 Determine whether or not the vector (8, 0, 5) is a linear combination of (1, 2, 3), (0, 1, 4), and (2, -1, 1). Solution

  4. Example 3 Can the vector (3, -4, -6) be a linear combination of (1, 2, 3), (-1, -1, -2), and (1, 4, 5). Solution

  5. Example 4 Express the vector (4, 5, 5) as a linear combination of the vectors (1, 2, 3), (-1, 1, 4), and (3, 3, 2). Solution

  6. Example 5 Determine whether the matrix is a linear combination of the matrices in the vector space M22 Solution

  7. Example 6 Determine whether the function is a linear combination of the functions and Solution

  8. Spanning Sets Definition The vectors v1, v2, …, vm are said to …………… a vector space if every vector in the space can be expressed as a …………………………. of these vectors.

  9. Example 7 Show that the vectors (1, 2, 0), (0, 1, -1), and (1, 1, 2) span R3. Solution

  10. Example 8 Show that the following matrices span the vector space M22 of 2  2 matrices. Solution

  11. 4.3 Linear Dependence and Independence • Definition • The set of vectors { v1, …, vm } in a vector space V is said to be …………..……if there exist scalars c1, …, cm …………, such that ……………….…… • The set of vectors { v1, …, vm } is ………………..…if ……..…………can only be satisfied when ………………

  12. Example 9 Show that the set {(1, 2, 0), (0, 1, -1), (1, 1, 2)} is linearly independent in R3. Solution

  13. Example 10 Show that the set of functions {x + 1, x – 1, – x + 5} is linearly dependent in P1. Solution

  14. Theorem 4.7 A set consisting of two or more vectors in a vector space is …………………it is possible to express ……… of them as a ………………………………………… Example 11 The set of vectors {(1, 2, 1) , (-1, -1, 0) , (0, 1,1)} is linearly ………………………………………………..… Example 12 The set of vectors {(2, -1, 3) , (4, -2, 6)} is linearly ………………………………………………..… Example 13 The set of vectors {(1, 2, 3) , (6, 5, 4)} is linearly ………………………………………………..…

  15. Theorem 4.8 Let V be a vector space. Any set of vectors in V that contains the…….is linearly …………. Example 14 The set of vectors {0,v1, v2, … , vn} is linearly……………………………………

  16. Theorem 4.9 Let the set {v1, …, vm} be linearly …………... in a vector space V. Any set of vectors in V ………………………. will …… be linearly ………………. Example 15 W={(1, 2, 3) , (2, 4, 6)} is linearly …………… U={(1, 2, 3) , (2, 4, 6), (4, 5, 6), (3, 5, 4)} is …………………… Note Let the set {v1, v2} be linearly independent, then {v1 + v2, v1 – v2} is also linearly………………………

  17. 4.4 Bases and Dimension • Definition • A finite set of vectors {v1, …, vm} is called a …………for a vector space V if: • the set ……………….. • the set ……………….... Standard Basis The set of n vectors …………………………………………… is a …….. for Rn. This basis is called the …………. basis for Rn. Theorem 4.11 Any two bases for a vector space V consist of the ………………..

  18. Definition If a vector space V has a basis consisting of n vectors, then the …………….. of V is said to be n and denoted by …………... Note Ch04_18

  19. Example 16 Prove that the set {(1, 0, -1), (1, 1, 1), (1, 2, 4)} is a basis for R3. Solution

  20. Example 17 Show that { f, g, h }, where f(x) = x2 + 1, g(x) = 3x – 1, and h(x) = –4x + 1 is a basis for P2. Solution

  21. Theorem 4.10 Let {v1, …, vn } be a basis for a vector space V. If {w1, …, wm} is a set of …………… vectors in V, then this set is linearly ………………. Example 18 Is the set {(1, 2), (-1, 3), (5, 2)} linearly independent inR2. Solution

  22. Theorem 4.14 • Let V be a vector space of dim(V)= n. • If S = {v1, …, vn} is a set of n ……………………. vectors in V S is a ………. for V. • If S = {v1, …, vn} is a set of n vectors V that …………… V S is a ………. for V.

  23. Example 19 Prove that the set B={(1, 3, -1), (2, 1, 0), (4, 2, 1)} is a basis for R3. Solution

  24. Example 20 State (with a brief explanation) whether the following statements are true or false. (a) The vectors (1, 2), (-1, 3), (5, 2) are linearly dependent in R2. (b) The vectors (1, 0, 0), (0, 2, 0), (1, 2, 0) span R3. (c) {(1, 0, 2), (0, 1, -3)} is a basis for the subspace of R3 consisting of vectors of the form (a, b, 2a-3b). (d) Any set of two vectors can be used to generate a two-dimensional subspace of R3. Note (b) dim(V)=n: *{v1, …, vn}Span V then it is linearly independent *Not linearly independent then not span. Solution

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