18 Orthogonal Complement #Orthogonalcomplement #orthogonal #complement

In this lecture, we explore the concept of the Orthogonal Complement in inner product spaces. We begin by drawing an analogy with the span of a subset—just as spans are always subspaces, so are orthogonal complements. These naturally occurring subspaces help us understand the structure of vector spaces more deeply.

We then discuss a powerful result: every vector in an inner product space can be uniquely expressed as the sum of one vector from a subspace and another from its orthogonal complement. This leads to the idea of orthogonal projection, and we prove that this projection is the closest point in the subspace to the original vector.

The video includes intuitive examples and closes with an important theorem: dim 𝑉 = dim 𝑊 + dim 𝑊⊥ showing how a space splits into orthogonal components.

📌 Topics Covered:
✔ Definition and Properties of Orthogonal Complement
✔ Subspace + Complement Analogy
✔ Unique Decomposition of Vectors
✔ Orthogonal Projections and Nearest Vectors
✔ Theorem: dim 𝑉 = dim 𝑊 + dim 𝑊⊥
✔ Solved Examples and Visual Insights

🎯 Ideal For:
– B.Sc. / M.Sc. Mathematics Students
– CSIR-NET, GATE, JAM Aspirants
– Anyone learning Inner Product Spaces or Linear Algebra in depth

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Видео 18 Orthogonal Complement #Orthogonalcomplement #orthogonal #complement канала The ClassRoom Study