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Consider a vector x in a linear vector space of dimension N spanned by basis {e^i}. You need to prove that x  has a unique expansion in terms of this basis. put these statements in order of proof.

Consider a vector x in a linear vector space of dimension N spanned by basis {e^i}. You need to prove that x  has a unique expansion in terms of this basis. put these statements in order of proof.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.2: Norms And Distance Functions
Problem 33EQ
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Consider a vector x in a linear vector space of dimension N spanned by basis {e^i}. You need to prove that x
 has a unique expansion in terms of this basis. put these statements in order of proof. 

Since the {ê; } forms a basis, these vectors are linearly independent. This contradicts our starting assumpion. Therefore any vector x in a vector space has a unique expansion in a given basis. Therefore we must have x; = y; for all i. N N 0 = Σ₁ x¡ê; - Σ₁ Viê¡ N 0 = Σ₁ (xi - Yi)ê¡
Transcribed Image Text:Since the {ê; } forms a basis, these vectors are linearly independent. This contradicts our starting assumpion. Therefore any vector x in a vector space has a unique expansion in a given basis. Therefore we must have x; = y; for all i. N N 0 = Σ₁ x¡ê; - Σ₁ Viê¡ N 0 = Σ₁ (xi - Yi)ê¡
0=X-X N N Since Σ1 x = Σvê; it follows i=1 i=1 that x₁ = y₁, Vi. From the axioms of a linear vector space it follows that x₁ = y; for all i. x; N Suppose x = Σ₁ x¡ê; and N X = Σyiê; where x; ‡ Y¡, Vi i=1
Transcribed Image Text:0=X-X N N Since Σ1 x = Σvê; it follows i=1 i=1 that x₁ = y₁, Vi. From the axioms of a linear vector space it follows that x₁ = y; for all i. x; N Suppose x = Σ₁ x¡ê; and N X = Σyiê; where x; ‡ Y¡, Vi i=1
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