linear algebra 3.3 Q11 **Chapter 3: Vector Spaces**Page 15610. Suppose \( \mathbf{v}_1, \ldots, \mathbf{v}_k \) are nonzero vectors with the property that \( \mathbf{v}_i \cdot \mathbf{v}_j = 0 \) whenever \( i \neq j \). Prove that \(\{ \mathbf{v}_1, \ldots, \mathbf{v}_k \} \) is linearly independent. (Hint: “Suppose \( c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \cdots + c_k \mathbf{v}_k = \mathbf{0}. \)” Start by showing \( c_1 = 0. \))

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Answered: 10. Suppose v1, ..., Vķ are nonzero…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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linear algebra 3.3 Q11

**Chapter 3: Vector Spaces**

Page 156

10. Suppose \( \mathbf{v}_1, \ldots, \mathbf{v}_k \) are nonzero vectors with the property that \( \mathbf{v}_i \cdot \mathbf{v}_j = 0 \) whenever \( i \neq j \). Prove that \(\{ \mathbf{v}_1, \ldots, \mathbf{v}_k \} \) is linearly independent. (Hint: “Suppose \( c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \cdots + c_k \mathbf{v}_k = \mathbf{0}. \)” Start by showing \( c_1 = 0. \))

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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