**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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Answered: 10. Suppose that U and V are finite…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

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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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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)