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10. Suppose that U and V are finite dimensional vector spaces and that F:VU is a linear map. prove that dim(V) = nullity (F) + rank (F)

10. Suppose that U and V are finite dimensional vector spaces and that F:VU is a linear map. prove that dim(V) = nullity (F) + rank (F)

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Problem Statement:** Consider vector spaces \( U \) and \( V \) that are finite-dimensional. Let \( F: V \rightarrow U \) be a linear map. Prove that: \[ \text{dim}(V) = \text{nullity}(F) + \text{rank}(F) \] This statement is a formulation of the Rank-Nullity Theorem, which relates the dimensionality of the domain of a linear transformation to its rank and nullity. The theorem asserts that the sum of the rank (the dimension of the image of the transformation) and the nullity (the dimension of the kernel of the transformation) equals the dimension of the domain vector space.
Transcribed Image Text:**Problem Statement:** Consider vector spaces \( U \) and \( V \) that are finite-dimensional. Let \( F: V \rightarrow U \) be a linear map. Prove that: \[ \text{dim}(V) = \text{nullity}(F) + \text{rank}(F) \] This statement is a formulation of the Rank-Nullity Theorem, which relates the dimensionality of the domain of a linear transformation to its rank and nullity. The theorem asserts that the sum of the rank (the dimension of the image of the transformation) and the nullity (the dimension of the kernel of the transformation) equals the dimension of the domain vector space.
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