Pure Mathematics C2: Vector Spaces

In this unit you will see how familiar properties of R2 and R3 can be generalised to other, very different sets of vectors through the concept of a vector space. Vector space is a mathematical structure that is one of the most important unifying concepts of pure mathematics.

72 flashcards · Mathematics · by lakesare

1. In mathematics, what is set VS space?

   In mathematics, a space is a set with some added structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself.Similarly, several types of numbers are in use (natural, integral, rational, real, complex); each one has its own definition; but just "number" is not used as a mathematical notion and has no definition.

2. How do we denote a Euclidean space?

   ℝn

3. [Distributivity] When is ∗ left-distributive over +? Given a set S, binary operation ∗, binary operation + on S

   x ∗ (y + z) = (x ∗ y) + (x ∗ z) When ∗ is commutative, the three conditions above are logically equivalent.

4. [Distributivity] When is ∗ right-distributive over +? Given a set S, binary operation ∗, binary operation + on S

   (y + z) ∗ x = (y ∗ x) + (z ∗ x) When ∗ is commutative, the three conditions above are logically equivalent.

5. [Distributivity] How do we read it?

   ∗ is left-distributive over +.

6. [Distributivity] How do we read it?

   ÷ is right-distributive over +. ÷ is not left-distributive over +.

7. [Polynomials] How do we denote this set of polynomials? where a, b, c ∈ R

   P3. The x occurring in a polynomial is commonly called a variable or an indeterminate. When the polynomial is considered as an expression, x is a fixed symbol which does not have any value (its value is "indeterminate"). However, when one considers the function defined by the polynomial, then x represents the argument of the function, and is therefore called a "variable". Many authors use these two words interchangeably.

8. [Polynomials] How can you simplify this notation?

9. [Polynomials] What kinds of polynomials does this set include?

   quadratic polynomials (2 is the highest present power) linear polynomials (1 is the highest present power) constants (0 is the highest present power) 0

10. Why is the real vector space called real?

    Because the scalars used in forming scalar multiples are real numbers.

11. Name 10 properties of the vector space (V, +, *).

    Vector addition (V, +) is an abelian group Scalar multiplication Closure Associativity Identity Distributivities Distributivity Distributivity

12. Visualize P3 VS quadratic polynomials.

13. [Vector spaces] Name the example vector from P3. Example of vector spaces: P3, ℂ, M23, R∞

    {4 + 3x + 77x2, ...}

14. [Vector spaces: set of complex numbers ℂ] Name the example vector from ℂ. Example of vector spaces: P3, ℂ, M23, R∞

    {3 + 2i, ...}

15. [Vector spaces: set M2,3 of 2×3 matrices with real entries] Name the example vector from M2,3. Example of vector spaces: P3, ℂ, M23, R∞

16. [Vector spaces: set R∞] Name the example vector from R∞. Example of vector spaces: P3, ℂ, M23, R∞

    {(2, 100, 8, .....................................................................), ...}

17. How are some sets of vectors special?

    Some sets of vectors can produce all other vectors.

18. What's linear combination VS linear equation?

    linear combination linear equation

19. Gauss-Jordan elimination VS row-reduction?

    It's the same, but usually we say: Gauss-Jordan Elimination - performed on the system of linear equations Row-reduction - performed on the matrix

20. Visually - what is common to all elements (vectors!) of a vector space?

    All of them have ordered positions, like a vector. {(2, 100, 8, .....................................................................), ...} {4 + 3x + 77x2, ...} {3 + 2i, ...}

21. Name 2 things that〈el〉can denote in mathematics.

    〈el〉- { all powers of el }. 〈S〉- { all linear combinations generated from S }

22. [Span] What does〈S〉look like visually?

23. What is the name of〈S〉?

    The span of S.

24. How do we denote the span of S?

    〈S〉

25. Visually imagine the〈S〉.

    Geometrically,〈S〉is the line y = x.

26. What's a good analogy for S and〈S〉?

    S is 2 hands of yours. 〈S〉is the area you get by swinging them around.

27. [Linear dependence] When is a set of vectors linearly dependent?

    When some v = linear combination of other vectors. Note that α1 = α2 = ··· = αk = 0 is a solution to the equation whether the set of vectors is linearly dependent or linearly independent. So the distinction between the two cases is whether there is a non-zero solution.

28. [Linear dependence] When do we say that vk is "dependent"?

    vk depends on other vectors when vk can be expressed via them.

29. [Linear dependence] What can you say about the linear dependence of { v1 }?

    { v1 } is linearly independent.

30. [Linear dependence] What can you say about the linear dependence of Ø?

    Ø is linearly independent.

31. When are: 2 points collinear VS 2 vectors collinear?

    2 points collinear - when they lie on the single line. 2 vectors collinear - vectors parallel to the same line.

32. [Linear dependence] Name 2 steps to test whether { v1, v2, ...vk } is linearly dependent?

    Write down unknown * v1 + ...unknown * vk = 0. If unknown = unknown = 0 is the only solution, then S is linearly independent.

33. [Linear dependence] How would you approach determining whether these a) VS b) are linearly dependent?

    a) Use a quick strategy of checking whether v1 is the multiple of v2. b) Use a longer strategy of writing out the system of linear equations.

34. [Linear dependence] Why did we decide to think about linear dependence?

    Because it lets us reason about the minimal set of vectors needed to span some vector space.

35. [Basis for a vector space] Name 2 conditions for a { v1, v2, ...vk } to be a basis of the vector space V?

    { v1, v2, ...vk } should span V { v1, v2, ...vk } should be linearly independent So that it's a minimal set of vectors.

36. [Basis for a vector space] What's the plural of basis?

    Bases.

37. [Basis for a vector space] Name a nice property that having some particular basis gives to every vector of our vector space V.

    We can express each vector of a vector space in a unique way.

38. [Basis for a vector space] To show S is a basis for some R3: show S is linearly independent - how? show S spans R3 - how?

    S is linearly independent we solve the system of equations to show that the only solution is unknown = unknown = 0. S spans R3 we solve the system of equations to show that any (x, y, z) in R3 can be expressed (in terms of vectors in S).

39. [Basis for a vector space] What is this called?

    The standard basis for Rn.

40. [Basis for a vector space] If the standard basis is so simple to use, why don't we always use it?

    Similarly, if we are considering a parallelogram, we may want to choose basis vectors along the sides of the parallelogram.

41. Why do we define a basis for a vector space (instead of defining it for any set of vectors)?

    Because〈S〉is always a vector space!

42. [Coordinates with respect to basis E] What's a short way to say "coordinates of v with respect to basis E"?

    E-coordinates of v.

43. [Coordinates with respect to basis E] How can we write v?

44. [Basis for a vector space] What's a short way to check whether {(), ...()} is a basis for our 5-dimensional vector space?

    We only need to check that our |basis| = n is linearly independent.

45. [Basis for a vector space] What's common between all bases of a "vector space V"?

    Each basis consists of the same n of vectors!

46. [Dimension of a vector space] Name 2 steps to find the dimension of, e.g., ℂ?

    Recall the standard basis of ℂ: { 1, i } Count the number of els in a basis: |{ 1, i }| = 2.

47. [Standard Basis] Visually imagine the standard basis of M2,2.

    The dimension of M2,2 is 4.

48. [Standard Basis] Visually imagine the standard basis of ℂ.

    The dimension of ℂ is 2.

49. [Standard Basis] Visually imagine the standard basis of Pn.

    The dimension of Pn is n.

50. [Standard Basis] Visually imagine the standard basis of Rn.

    The dimension of Rn is n.

51. [Dimension] What's the dimension of M2,2? Standard bases

    |basis| = 4.

52. [Dimension] What's the dimension of ℂ? Standard bases

    |basis| = 2.

53. [Basis for a vector space] Imagine visually the basis of a vector space.

54. [Basis for a vector space] Imagine visually the dimension of a vector space.

    The number of vectors in a basis.

55. [Subspace] When is some set a subspace of the vector space S?

    When: it's a subset it's a vector space, too

56. [Subspace] Name 3 points we need to check to show S ⊆ V is a subspace? Knowing V is a vector space, and S is a subset of V.

    0 ∈ S S is closed under vector addition S is closed under scalar multiplication.

57. [Comparison with groups] Groups - subgroups Vector spaces - ____________?

    Subspaces

58. [Comparison with groups] What's common between〈el〉and〈S〉? 〈el〉- { all powers of el }. 〈S〉- { all linear combinations generated from S }

    〈el〉always creates a subgroup of the Group, 〈S〉always creates a subspace of the Vector Space.

59. How would we approach this?

    We write (x, y, z, x + 2y – z) in terms of x(1, 0, 0, 1) + y(0, 1, 0, 2) + z(0, 0, 1, –1).

60. [Orthogonality] Algebraically, when are the 2 vectors orthogonal? Geometrically, this means that the vectors v1 and v2 are at right angles to each other.

    When v1 · v2 = 0.

61. [Orthogonality] When is the set of vectors in Rn orthogonal set?

    When all vectors are pairwise orthogonal.

62. [Orthogonality] Visually imagine why orthogonal set → linearly independent set.

63. [Orthogonality] Why is it nice to have an orthogonal basis?

    Because it's particularly easy to convert: vstandard basis → vorthogonal basis. Otherwise we'd need to solve a system of linear equations.

64. [Orthogonality] What does Gram-Schmidt orthogonalisation process let us do? Erhard Schmidt Jørgen Pedersen Gram

    It lets us: knowing any basis → derive orthogonal basis.

65. [Orthogonality] What's an orthogonal basis?

    A basis whose vectors are pairwise orthogonal.

66. Gram-Schmidt _______________ process?

    Gram-Schmidt orthogonalisation process

67. [Orthogonality] How would you approach finding the orthogonal basis if we already know some basis?

    Use the Gram-Schmidt orthogonalisation process.

68. [Orthogonality] Orthogonal basis VS orthonormal basis?

    Orthogonal basis - all vectors are pairwise orthogonal. Orthonormal basis - orthogonal + |v ⃗| = 1.

69. [Magnitude and dot product] Imagine visually why magnitude of v = √v · v.

    Because |v| = √v12 + v22, etc.

70. [Orthonormality] How does this theorem become simpler if we have an orthonormal basis (instead of the orthogonal)?

71. [Orthonormality] How do we find the orthonormal basis from the orthogonal basis?

    Simply divide each vector by its magnitude! As a shorthand for 'scalar multiply a vector by the reciprocal of its magnitude', we may say 'divide a vector by its magnitude'.

72. [Span] What do we ultimately want to show if we want to show that S spans A?

    Equality of sets: <S> = A.