Linear Algebra Lecture 76 | Induced Map T̃: Isomorphism from V/null T to Range(T) (NET/GATE/JAM)

In this lecture (Lecture 76), we study the induced map from the quotient space V/null T to the range of T. This is one of the most beautiful ideas in Linear Algebra and forms the heart of the First Isomorphism Theorem. ⸻ ⭐ Topics Covered in this Lecture 1. Definition of the Induced Map T̃ We define a new function: T̃(v + null T) = T(v) This makes sense because T gives the same value for every vector of the same coset v + null T. ⸻ 2. T̃ maps cosets to the range of T The induced function naturally lands in range(T), since T(v) always belongs to it. ⸻ 3. T̃ is an Isomorphism We state (and motivate) the fundamental fact: The induced map T̃ is an isomorphism from V/null T onto range(T). This is the structural core of the First Isomorphism Theorem in linear algebra. In the next lecture, we will: • prove linearity • check well-definedness • find the kernel and image • and finish the complete proof of the isomorphism ⸻ 🔍 Ideal for: • MSc Mathematics • CSIR-UGC NET Mathematics • GATE Mathematics (MA) • NBHM Exam Preparation • IIT JAM Mathematical Sciences • BSc Mathematics (Advanced Linear Algebra) • Mathematics Faculty and Researchers • Students following Axler’s Linear Algebra Done Right