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(7) Let V and U be vector spaces and W a subspace of V. Suppose T : V → U is a linear transformation with ker(T) = W and define T : V \W →U by T(v + W) := T(v). Prove T is an isomorphism from V \ W to Im(T), that is V \ ker(T) ~ Im(T). (In abstract algebra, this is known as the first isomorphism theorem!)

(7) Let V and U be vector spaces and W a subspace of V. Suppose T : V → U is a linear transformation with ker(T) = W and define T : V \W →U by T(v + W) := T(v). Prove T is an isomorphism from V \ W to Im(T), that is V \ ker(T) ~ Im(T). (In abstract algebra, this is known as the first isomorphism theorem!)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(7) Let ? and ? be vector spaces and ? a subspace of ? . Suppose ? : ? → ? is a linear ̃︀ ̃︀ transformation with ???(?) = ? and define ? : ? ∖ ? → ? by ?(? + ?) := ?(?). ̃︀ Prove ? is an isomorphism from ? ∖ ? to ??(?), that is ? ∖ ???(?) ≃ ??(?). (In abstract algebra, this is known as the first isomorphism theorem!)

(7) Let \( V \) and \( U \) be vector spaces and \( W \) a subspace of \( V \). Suppose \( T: V \to U \) is a linear transformation with \( \text{ker}(T) = W \) and define \( \widetilde{T}: V / W \to U \) by \( \widetilde{T}(v + W) := T(v) \). Prove \( \widetilde{T} \) is an isomorphism from \( V / W \) to \( \text{Im}(T) \), that is \( V / \text{ker}(T) \cong \text{Im}(T) \). (In abstract algebra, this is known as the first isomorphism theorem!)
Transcribed Image Text:(7) Let \( V \) and \( U \) be vector spaces and \( W \) a subspace of \( V \). Suppose \( T: V \to U \) is a linear transformation with \( \text{ker}(T) = W \) and define \( \widetilde{T}: V / W \to U \) by \( \widetilde{T}(v + W) := T(v) \). Prove \( \widetilde{T} \) is an isomorphism from \( V / W \) to \( \text{Im}(T) \), that is \( V / \text{ker}(T) \cong \text{Im}(T) \). (In abstract algebra, this is known as the first isomorphism theorem!)
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