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dadp + ... + TAip + IAp = A pue ... v = c\V1 +C2V2+ + CpVp = V means that any given vector in V is a linear Hint: The fact that Span(S) V any in V is a linear combination of the in S. So let v be some given in V. is any v e V be as a of left to is if there are and that

dadp + ... + TAip + IAp = A pue ... v = c\V1 +C2V2+ + CpVp = V means that any given vector in V is a linear Hint: The fact that Span(S) V any in V is a linear combination of the in S. So let v be some given in V. is any v e V be as a of left to is if there are and that

Advanced Engineering Mathematics
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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The image contains text explaining concepts related to vector spaces in linear algebra. Here is a transcription and explanation adapted for educational purposes: --- To determine if a set \( S \) is closed under addition and scalar multiplication, it must be shown that \( S \) is a basis for \( V \). This is because a basis is a set of vectors in a vector space that is linearly independent and spans the entire space, meaning every vector \( v \) in \( V \) can be expressed uniquely as a linear combination of the vectors in \( S \). 1. **Closed under Addition and Scalar Multiplication:** - Prove that if \( S \) is a set of vectors such that \( \text{span}(S) = V \), then any vector \( v \) can be uniquely expressed as a linear combination of the vectors in \( S \). 2. **Linear Combinations:** - Let the vectors in \( S \) be \( v_1, v_2, \ldots, v_p \). - The fact that \( \text{span}(S) = V \) means any vector in \( V \) is a linear combination of vectors in \( S \). Equations given: - \( v = c_1v_1 + c_2v_2 + \cdots + c_pv_p \) - \( v = d_1v_1 + d_2v_2 + \cdots + d_pv_p \) Where \( c_1, c_2, \ldots, c_p \) and \( d_1, d_2, \ldots, d_p \) are scalars, such that there exist unique scalars for any vector \( v \) expressed as a linear combination of \( v_1, v_2, \ldots, v_p \). **Graph or Diagram Explanation:** There are no graphs or diagrams in this text. --- This content aims to help students understand the foundational aspects of vector spaces and the characteristics of a basis in linear algebra.
Transcribed Image Text:The image contains text explaining concepts related to vector spaces in linear algebra. Here is a transcription and explanation adapted for educational purposes: --- To determine if a set \( S \) is closed under addition and scalar multiplication, it must be shown that \( S \) is a basis for \( V \). This is because a basis is a set of vectors in a vector space that is linearly independent and spans the entire space, meaning every vector \( v \) in \( V \) can be expressed uniquely as a linear combination of the vectors in \( S \). 1. **Closed under Addition and Scalar Multiplication:** - Prove that if \( S \) is a set of vectors such that \( \text{span}(S) = V \), then any vector \( v \) can be uniquely expressed as a linear combination of the vectors in \( S \). 2. **Linear Combinations:** - Let the vectors in \( S \) be \( v_1, v_2, \ldots, v_p \). - The fact that \( \text{span}(S) = V \) means any vector in \( V \) is a linear combination of vectors in \( S \). Equations given: - \( v = c_1v_1 + c_2v_2 + \cdots + c_pv_p \) - \( v = d_1v_1 + d_2v_2 + \cdots + d_pv_p \) Where \( c_1, c_2, \ldots, c_p \) and \( d_1, d_2, \ldots, d_p \) are scalars, such that there exist unique scalars for any vector \( v \) expressed as a linear combination of \( v_1, v_2, \ldots, v_p \). **Graph or Diagram Explanation:** There are no graphs or diagrams in this text. --- This content aims to help students understand the foundational aspects of vector spaces and the characteristics of a basis in linear algebra.
The visible text in the image is: "then \( c_1 = d_1, c_2 = d_2, ..., c = d_p \). at \( \mathbb{U} \). So \( \mathbb{D} \) and let S be the set of vectors (actu" There are no graphs or diagrams visible in this image.
Transcribed Image Text:The visible text in the image is: "then \( c_1 = d_1, c_2 = d_2, ..., c = d_p \). at \( \mathbb{U} \). So \( \mathbb{D} \) and let S be the set of vectors (actu" There are no graphs or diagrams visible in this image.
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