Linear Algebra: Permutations and Subspaces
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Questions and Answers

What is the result of the expression (123)(45) - (45)(123)?

  • Another constant value
  • The product of 123 and 123
  • 0 (correct)
  • 123 × 45

    • Which of the following correctly describes a basis of a vector space?

  • Vectors that are all linearly dependent
  • A collection of vectors that includes at least one zero vector
  • Any set of vectors that can span the space
  • A set of vectors that covers the entire space without redundancy (correct)

    • What is the definition of linearly independent vectors?

  • A set of vectors that does not have the zero vector
  • Vectors that can be written as a linear combination of one another
  • Vectors that can form a basis for some vector space (correct)
  • Vectors that only span a small subset of the space

    • If a set of vectors is linearly dependent, which of the following must be true?

    <p>At least one vector in the set can be expressed as a combination of others</p> Signup and view all the answers

    Consider the set of vectors {v1, v2, v3}. Which condition must hold for them to be termed a sub-space?

    <p>The set must be closed under vector addition and scalar multiplication</p> Signup and view all the answers

    Study Notes

    Proving the Equation

    • The expression involves the product of permutations (123) and (45)
    • (123) represents the cycle that sends 1 to 2, 2 to 3, and 3 to 1
    • (45) represents the cycle that sends 4 to 5 and 5 to 4
    • The product of permutations is calculated by applying the cycles consecutively
    • (123)(45) sends 1 to 2, 2 to 3, 3 to 1, 4 to 5, and 5 to 4
    • (45)(123) sends 1 to 2, 2 to 3, 3 to 1, 4 to 5, and 5 to 4
    • Therefore, (123)(45) and (45)(123) represent the same permutation resulting in 0 when subtracted
    • This illustrates that the order of applying cycles within a product can affect the resulting permutation

    Sub-Space

    • A sub-space of a vector space is a subset that is also a vector space under the same operations
    • For example, the set of all vectors in R^3 with the form (x, 0, 0) is a sub-space of R^3 because it is closed under addition and scalar multiplication

    Basis of a Vector Space

    • A basis of a vector space is a linearly independent set that spans the entire vector space
    • For example, the set {(1, 0), (0, 1)} is a basis for R^2 because any vector in R^2 can be written as a linear combination of these two vectors, and they are linearly independent

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    Description

    This quiz covers key concepts in linear algebra, focusing on permutations, subspaces, and the basis of vector spaces. Learn how the product of permutations can yield different results based on the order of application, and explore the criteria for identifying a subspace in vector spaces. Test your understanding of these fundamental topics in linear algebra.

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