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Questions and Answers
What is the result of the expression (123)(45) - (45)(123)?
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?
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?
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?
If a set of vectors is linearly dependent, which of the following must be true?
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Consider the set of vectors {v1, v2, v3}. Which condition must hold for them to be termed a sub-space?
Consider the set of vectors {v1, v2, v3}. Which condition must hold for them to be termed a sub-space?
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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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