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be a linear transformation from a vector space V into a
Chapter 4, Problem 30E(choose chapter or problem)
be a linear transformation from a vector space V into a vector space W . Prove that the range of T is a subspace of W . [Hint: Typical elements of the range have the form
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QUESTION:
be a linear transformation from a vector space V into a vector space W . Prove that the range of T is a subspace of W . [Hint: Typical elements of the range have the form
ANSWER:Solution 30EStep 1 of 2Suppose is a linear transformation, where are vector spaces.The range of T is defined as .The objective is to prove that is a subspace of W.Since T is a linear transformation and , so the zero vector of W is in .Let Consider Since T is a linear transformationSince and V is a vector space, so Thus, Therefore, is closed under vector addition.