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Section 4.1 Vectors in ℝ n

Section 4.1 Vectors in ℝ n. ℝ n Vectors Vector addition Scalar multiplication. Def. Let v 1 , v 2 , . . . v n be vectors in ℝ n . A linear combination of these vectors is an expression c 1 v 1 + c 2 v 2 + . . . + c n v n where c 1 , c 2 , . . . . , c n are some scalars.

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Section 4.1 Vectors in ℝ n

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  1. Section 4.1Vectors in ℝn

  2. ℝn Vectors Vector addition Scalar multiplication.

  3. Def. Let v1,v2, . . . vn be vectors in ℝn. A linear combination of these vectors is an expression c1v1 +c2v2 + . . . + cnvn where c1, c2, . . . . , cn are some scalars.

  4. Ex. (a) If possible, write <6, 7> as a linear combination of <2, 3> and <1, 1>.

  5. Ex. (b) If possible, write <10, 4> as a linear combination of <5, 2> and <3, 1>.

  6. Ex. (c) If possible, write <7, 10> as a linear combination of <1, 2> and <3, 6>.

  7. Ex. Verify your answers geometrically for the last example. (a) <6, 7> written as a linear combination of <2, 3> and <1, 1>.

  8. Ex. Verify your answers geometrically for the last example. (b) <10, 4> written as a linear combination of <5, 2> and <3, 1>.

  9. Ex. Verify your answers geometrically for the last example. (c) <7, 10> written as a linear combination of <1, 2> and <3, 6>.

  10. Properties of vectors in ℝn Let u, v, and w be vectors in ℝn and let c and d be scalars. 1. u + vis in ℝn 2. u + v= v + u 3. (u + v) + w = u + (v + w) 4. There is a vector 0 such that u + 0 = u for all u in ℝn 5. For all u in ℝn there is a vector –u such that u + (–u) = 0 6. cu is in ℝn 7. c(u + v) = cu + cv 8. (c + d)u = cu + du 9. c(du) = (cd)u 10. 1(u) = u

  11. Section 4.2Vector Spaces

  12. Def. Let V be a set on which two operations (called vector addition and scalar multiplication) are defined. V is called a vector space if V satisfies the following properties. For any u, v, and w be vectors in V and any scalars c and d: 1. u + vis in V 2. u + v= v + u 3. (u + v) + w = u + (v + w) 4. There is a vector 0 such that u + 0 = u for all u in V 5. For all u in V there is a vector –u such that u + (–u) = 0 6. cu is in V 7. c(u + v) = cu + cv 8. (c + d)u = cu + du 9. c(du) = (cd)u 10. 1(u) = u

  13. Examples of various vector spaces (each with the standard operations of addition and scalar multiplication): 1. ℝn – the set of all n–tuples.

  14. Examples of various vector spaces (each with the standard operations of addition and scalar multiplication): 2. C(–∞,∞) – the set of all continuous functions defined on the real line.

  15. Examples of various vector spaces (each with the standard operations of addition and scalar multiplication): 3. C[a, b] – the set of all continuous functions defined on a closed interval [a, b].

  16. Examples of various vector spaces (each with the standard operations of addition and scalar multiplication): 4. P – the set of all polynomials.

  17. Examples of various vector spaces (each with the standard operations of addition and scalar multiplication): 5. Pn – the set of all polynomials of degree ≤ n.

  18. Examples of various vector spaces (each with the standard operations of addition and scalar multiplication): 6. Mm,n – the set of all mxn matrices.

  19. Ex. Show that the following are not examples of vector spaces. (a) The set of all integers (with the standard operations).

  20. Ex. Show that the following are not examples of vector spaces. (b) The set of all polynomials with degree 2 (with the standard operations).

  21. Ex. Show that the following are not examples of vector spaces. (c) The set of all invertible 2x2 matrices (with the standard operations).

  22. Ex. Show that the following are not examples of vector spaces. (d) The set of all matrices (with the standard operations).

  23. Ex. Let V = ℝ2 with the standard operation of addition and the following nonstandard definition of scalar multiplication: c (x, y) = (cx, 0). Determine whether or not V is a vector space.

  24. Section 4.3Subspaces of Vector Spaces

  25. Def. Let V be a vector space and let W be a subset of V. If W is itself a vector space then W is said to be a subspace of V. Note: Every vector space contains at least two subspaces, the vector space itself and the set consisting only of the zero element. These are called the trivial subspaces.

  26. Ex. Determine whether or not the following are subspaces. (a) Let V = ℝ3 with the standard operations. Define W to be { (x,y,0) : x and y are real numbers}.

  27. Ex. Determine whether or not the following are subspaces. (b) Let V = ℝ3 with the standard operations. Define W to be { (x,y,1) : x and y are real numbers}.

  28. Theorem: Test for a Subspace Let V be a vector space and suppose W is a subset of V. W is a subspace of V if and only if the following hold: 1. If u and v are in W then u +v is in W. 2. If u is in W then cu is in W.

  29. Ex. Let a, b, and c be constants. Show that the set of all solutions to the differential equation ay″ + by′ + cy = 0 is a subspace (a subspace of the vector space of all functions).

  30. Note: W cannot be a subspace of V unless W contains the zero element.

  31. Ex. Which of the following are subspaces? (a)V = ℝ2 . W = the solution set of 2x + 3y = 5.

  32. Ex. Which of the following are subspaces? (b)V = ℝ2 . W = the solution set of -5x + y = 0.

  33. Ex. Which of the following are subspaces? (c)V = ℝ3 . W = the solution set of 3x + 8y – 4z = 7.

  34. Ex. Which of the following are subspaces? (d)V = ℝ3 . W = the solution set of -2x + 12y – z = 0.

  35. All nontrivial subspaces of ℝ2 are lines which pass through the origin. All nontrivial subspaces of ℝ3 are planes which pass through the origin or lines which pass through the origin.

  36. Ex. Is W = {(x, x+ z, z) : x and z are real numbers} a subspace of ℝ3?

  37. Ex. Show that the solution set to x2 + y2 = 1 is not a subspace of ℝ2.

  38. Ex. (a) Let V = ℝ3 with the standard operations. Define W to be {(x,y,0) : x and y are real numbers}. What can W be identified with?

  39. Ex. (b) Let V = ℝ3 with the standard operations. Define W to be {(t, t, 3t) : t is a real number}. What can W be identified with?

  40. Notation: P0, P1, P2, P3, . . . . , P

  41. Section 4.4Spanning Sets and Linear Independence

  42. Recall: Let v1,v2, . . . vn be vectors in ℝn. A linear combination of these vectors is an expression c1v1 +c2v2 + . . . + cnvn where c1, c2, . . . . , cn are some scalars.

  43. Ex. (a) If possible, write f (x) = x2 + x + 1 as a linear combination of g1(x) = x2 + 2x + 3, g2(x) = x + 2, and g3(x) = –x2 + 1.

  44. Ex. (b) If possible, write as a linear combination of , , and

  45. Def. Let S = {v1,v2, . . . vn} be a subset of a vector space V. The span of S is the set of all linear combinations of elements of S. span(S) = {c1v1 +c2v2 + . . . + cnvn : c1, c2, . . . . cn are scalars}

  46. Def. Let S = {v1,v2, . . . vn} be a subset of a vector space V. If every vector of V can be written as a linear combination of elements of S then we say S spans V. Put another way, S spans V if span(S) = V.

  47. Ex. Show that {(1, 0, 0), (0, 1, 0), (0, 0, 1)} spans ℝ3.

  48. Ex. Does {(1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 1)} span ℝ3?

  49. Ex. Does {(1, 0, 0), (0, 1, 0)} span ℝ3?

  50. Ex. Determine whether or not {(1, 2, 3), (4, 5, 6), (7, 8, 9)} spans ℝ3. If it does, then write a general vector (x, y, z) as a linear combination of these vectors. If it does not, then find a vector in ℝ3 which it does not span.

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