Linear Algebra: Permutations and Subspaces

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

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

What is the definition of linearly independent vectors?

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

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

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

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.