Math Problem Statement
Show that 1x = x for all in F^n.
Solution
To prove that for all in , where denotes the vector space of -tuples over a field , we proceed as follows:
Proof:
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Vector Space Definition: is defined as the set of all -tuples , where each is an element of the field . The vector space is equipped with two operations: vector addition and scalar multiplication.
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Scalar Multiplication: Scalar multiplication in is defined as follows: for any scalar and any vector , where each is the multiplication of the scalar with the component in the field .
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Multiplicative Identity: The field contains a multiplicative identity element, denoted by 1, such that for any , This property applies to each component of the vector .
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Applying Scalar Multiplication with 1: Let be an arbitrary vector in . Then, multiplying this vector by the scalar 1 gives: Using the property of the multiplicative identity in the field , Therefore,
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Conclusion: We have shown that for any vector in the vector space , multiplying by the scalar 1 leaves the vector unchanged, i.e., . This completes the proof.
Would you like more details or have any questions?
Here are 5 related questions to consider:
- What is the difference between a field and a vector space ?
- How does scalar multiplication in relate to matrix multiplication?
- Can you prove that for all ?
- What role does the distributive property play in vector spaces?
- How can the concept of basis vectors be used to understand operations in ?
Tip: Understanding the interaction between scalars and vectors is crucial when working with vector spaces, especially when dealing with transformations and linear mappings.
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Math Problem Analysis
Mathematical Concepts
Vector Spaces
Scalar Multiplication
Field Theory
Formulas
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Theorems
Multiplicative Identity
Suitable Grade Level
Advanced Undergraduate