linear algebra 3.3 Q11 **Exercise 11**Suppose \( \mathbf{v_1}, \ldots, \mathbf{v_n} \) are nonzero, mutually orthogonal vectors in \( \mathbb{R}^n \).a. Prove that they form a basis for \( \mathbb{R}^n \). (Use Exercise 10.)b. Given any \( \mathbf{x} \in \mathbb{R}^n \), give an explicit formula for the coordinates of \( \mathbf{x} \) with respect to the basis \(\{\mathbf{v_1}, \ldots, \mathbf{v_n}\}\).c. Deduce from your answer to part b that \( \mathbf{x} = \sum_{i=1}^{n} \text{proj}_{\mathbf{v}_i} \mathbf{x} \).

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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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linear algebra 3.3 Q11

**Exercise 11**

Suppose \( \mathbf{v_1}, \ldots, \mathbf{v_n} \) are nonzero, mutually orthogonal vectors in \( \mathbb{R}^n \).

a. Prove that they form a basis for \( \mathbb{R}^n \). (Use Exercise 10.)

b. Given any \( \mathbf{x} \in \mathbb{R}^n \), give an explicit formula for the coordinates of \( \mathbf{x} \) with respect to the basis \(\{\mathbf{v_1}, \ldots, \mathbf{v_n}\}\).

c. Deduce from your answer to part b that \( \mathbf{x} = \sum_{i=1}^{n} \text{proj}_{\mathbf{v}_i} \mathbf{x} \).

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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