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Tet U= Span (1) a) Find a basis of U. Let 1 H 00 le a subspace of IR3 ) •Using the basis you found in () conate a morta Band use this mortria, to find the coordinate vertort, of x interms of subspace of

Tet U= Span (1) a) Find a basis of U. Let 1 H 00 le a subspace of IR3 ) •Using the basis you found in () conate a morta Band use this mortria, to find the coordinate vertort, of x interms of subspace of

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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**Title: Linear Algebra: Finding a Basis and Coordinate Vectors** **Topic: Vector Spaces and Subspaces** **Let \( U = \text{span} \left( \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \right) \) be a subspace of \(\mathbb{R}^3\).** ### a) Find a basis of \( U \). The problem asks us to determine a set of vectors that can form a basis for the subspace \( U \). A basis is a set of linearly independent vectors that span the vector space or subspace. ### b) Let \( x = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \). - **Task 1:** Use the basis you found in (a) to create a matrix \( B \). - **Task 2:** Use this matrix to find the coordinate vector of \( x \) in terms of the subspace \( U \). The aim is to express the vector \( x \) in terms of the basis identified for the subspace \( U \). This involves using techniques from linear algebra to find the coordinate equivalents in the basis \( B \).
Transcribed Image Text:**Title: Linear Algebra: Finding a Basis and Coordinate Vectors** **Topic: Vector Spaces and Subspaces** **Let \( U = \text{span} \left( \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \right) \) be a subspace of \(\mathbb{R}^3\).** ### a) Find a basis of \( U \). The problem asks us to determine a set of vectors that can form a basis for the subspace \( U \). A basis is a set of linearly independent vectors that span the vector space or subspace. ### b) Let \( x = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \). - **Task 1:** Use the basis you found in (a) to create a matrix \( B \). - **Task 2:** Use this matrix to find the coordinate vector of \( x \) in terms of the subspace \( U \). The aim is to express the vector \( x \) in terms of the basis identified for the subspace \( U \). This involves using techniques from linear algebra to find the coordinate equivalents in the basis \( B \).
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