Prove that 𝐯-𝐰 ≥|𝐯-𝐰|. Interpret this result pictorially. |...

Name: Problem 39, Chapter 3: Applied Linear Algebra (Undergraduate Texts in Mathematics)
Uploaded: 2021-08-07T20:07:28-08:00
Duration: 2 min 39 s
Channel: Ziya Ogron
Description: Prove that 𝐯-𝐰 ≥|𝐯-𝐰|. Interpret this result pictorially.

Key Concepts

Norm

A norm is a function that assigns a non-negative length or size to vectors in a vector space. It satisfies properties such as positivity, scalar multiplication, and the triangle inequality. Norms provide a measure of distance from the origin, and they are foundational in understanding concepts like convergence, continuity, and geometric relationships in spaces.

Triangle Inequality

The triangle inequality is a fundamental property of norms stating that the norm of the sum of two vectors is less than or equal to the sum of their norms. This reflects the intuitive idea that, in a geometric space, the direct path between two points is the shortest. It is the basis for many results in analysis and geometry, including the reverse triangle inequality.

Reverse Triangle Inequality

The reverse triangle inequality is a consequence of the triangle inequality. It states that the norm of the difference between two vectors is at least the absolute difference of their norms. This inequality provides a lower bound on the distance between two vectors, ensuring that removing one vector from another cannot shrink the distance by more than the size of the vector being removed.

Geometric Interpretation

Pictorially, the reverse triangle inequality can be understood by visualizing vectors as points in space. Think of two circles centered at the origin with radii equal to the norms of the vectors. The minimum distance between any point on one circle and any point on the other circle is the difference of the radii. This illustrates that the difference in lengths of the vectors provides a lower bound on the distance between them.

Transcript

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