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Let V and V' be vector spaces over a field F. Then for any t1, t2 ∈ Hom (V, V')
- a)ρ(λ t1) = λ ρ(t1) ∀ λ ≠ 0 ∈ F
- b)ρ(t1) - ρ(t2)| ≥ ρ(t1 + t2)
- c)Both are true.
- d)None of these.
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
Let V and V be vector spaces over a field F. Then for any t1, t2∈...
A function ρ: V → V' is a linear transformation if it satisfies the following properties for all vectors u and v in V and all scalars λ in the underlying field F:
⇒ ρ(u + v) = ρ(u) + ρ(v) (Preservation of vector addition)
⇒ ρ(λu) = λ ρ(u) (Preservation of scalar multiplication)
⇒ ρ(λu) = λ ρ(u) (Preservation of scalar multiplication)
Most Upvoted Answer
Let V and V be vector spaces over a field F. Then for any t1, t2∈...
Understanding the Homomorphism Property
In the context of vector spaces, the statement pertains to properties of a homomorphism, specifically in terms of a norm or a functional. Let's break down the details of option 'A'.
Property of Homomorphism
- Given t1, t2 in Hom(V, V), we analyze the claim:
- ρ(λt1) = λρ(t1) for all λ ≠ 0 in F.
- This property indicates that the function ρ is *homogeneous* of degree 1. This means that scaling the input (a linear transformation in this case) scales the output by the same factor.
Significance of Homogeneity
- Homogeneity is a crucial aspect of linear transformations and ensures that the transformation behaves predictably under scalar multiplication.
- This property is essential in various mathematical contexts, including linear algebra and functional analysis, confirming that ρ respects the structure of the vector space.
Evaluation of Other Options
- Option b states the triangle inequality: ρ(t1) - ρ(t2) ≥ ρ(t1 + t2). This is not universally valid for all norms or functionals unless specified.
- Option c states both conditions are true; however, since option b is not necessarily valid, option c fails as well.
Conclusion
- Hence, the correct answer is option 'A', which confirms the homogeneity property of the homomorphism ρ is indeed valid for all linear transformations t1, t2 in the vector space V. This highlights the consistency and reliability of linear transformations in vector spaces over a field F.
In the context of vector spaces, the statement pertains to properties of a homomorphism, specifically in terms of a norm or a functional. Let's break down the details of option 'A'.
Property of Homomorphism
- Given t1, t2 in Hom(V, V), we analyze the claim:
- ρ(λt1) = λρ(t1) for all λ ≠ 0 in F.
- This property indicates that the function ρ is *homogeneous* of degree 1. This means that scaling the input (a linear transformation in this case) scales the output by the same factor.
Significance of Homogeneity
- Homogeneity is a crucial aspect of linear transformations and ensures that the transformation behaves predictably under scalar multiplication.
- This property is essential in various mathematical contexts, including linear algebra and functional analysis, confirming that ρ respects the structure of the vector space.
Evaluation of Other Options
- Option b states the triangle inequality: ρ(t1) - ρ(t2) ≥ ρ(t1 + t2). This is not universally valid for all norms or functionals unless specified.
- Option c states both conditions are true; however, since option b is not necessarily valid, option c fails as well.
Conclusion
- Hence, the correct answer is option 'A', which confirms the homogeneity property of the homomorphism ρ is indeed valid for all linear transformations t1, t2 in the vector space V. This highlights the consistency and reliability of linear transformations in vector spaces over a field F.
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