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Give a basis for span(S), where S is the set given below. 1 1 3 -1 -1 -2 1 1 0 2 2 4 Number of Vectors: 1 [[0]) 0 -2 5 -8 -8

Give a basis for span(S), where S is the set given below. 1 1 3 -1 -1 -2 1 1 0 2 2 4 Number of Vectors: 1 [[0]) 0 -2 5 -8 -8

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Topic: Linear Algebra - Basis and Span** **Problem Statement:** "Give a basis for \( \text{span}(S) \), where \( S \) is the set given below." The set \( S \) is composed of the following four vectors: \[ \left\{ \begin{bmatrix} 1 \\ -1 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ -1 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ -2 \\ 0 \\ 4 \end{bmatrix}, \begin{bmatrix} -2 \\ 5 \\ -8 \\ -8 \end{bmatrix} \right\} \] **Analysis:** * The objective is to determine a basis for the span of the set \( S \). * The basis is a set of vectors that spans the vector space and is linearly independent. **Result:** Underneath the problem, you will find: - **Number of Vectors:** 1 - The resulting basis is represented by the zero vector: \[ \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \] **Explanation:** The zero vector suggests that the vectors in \( S \) do not span a subspace other than the trivial subspace (zero vector), indicating linear dependence within the set. In this context, it seems that the given vectors, when analyzed, did not provide any linearly independent vectors to form a basis other than the trivial solution.
Transcribed Image Text:**Topic: Linear Algebra - Basis and Span** **Problem Statement:** "Give a basis for \( \text{span}(S) \), where \( S \) is the set given below." The set \( S \) is composed of the following four vectors: \[ \left\{ \begin{bmatrix} 1 \\ -1 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ -1 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ -2 \\ 0 \\ 4 \end{bmatrix}, \begin{bmatrix} -2 \\ 5 \\ -8 \\ -8 \end{bmatrix} \right\} \] **Analysis:** * The objective is to determine a basis for the span of the set \( S \). * The basis is a set of vectors that spans the vector space and is linearly independent. **Result:** Underneath the problem, you will find: - **Number of Vectors:** 1 - The resulting basis is represented by the zero vector: \[ \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \] **Explanation:** The zero vector suggests that the vectors in \( S \) do not span a subspace other than the trivial subspace (zero vector), indicating linear dependence within the set. In this context, it seems that the given vectors, when analyzed, did not provide any linearly independent vectors to form a basis other than the trivial solution.
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