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orthogonal to every vector in a basis in Null(A¹). (c) Use the Gram-Schmidt process to find an orthogonal basis for W.

orthogonal to every vector in a basis in Null(A¹). (c) Use the Gram-Schmidt process to find an orthogonal basis for W.

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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2. **Problem 2** Let \( A = \begin{pmatrix} 2 & 1 & 1 \\ 1 & -3 & 0 \\ -1 & -1 & 1 \\ 1 & 0 & -1 \end{pmatrix} \), \( W = \text{col}(A) \), and \( x = \begin{pmatrix} 3 \\ -4 \\ -1 \\ 2 \end{pmatrix} \). (a) Find \( \text{Proj}_W(x) \). (b) Construct bases for \( W \) and \( \text{Null}(A^\top) \) and verify that every vector in a basis for \( W \) is orthogonal to every vector in a basis in \( \text{Null}(A^\top) \). (c) Use the Gram-Schmidt process to find an orthogonal basis for \( W \).
Transcribed Image Text:2. **Problem 2** Let \( A = \begin{pmatrix} 2 & 1 & 1 \\ 1 & -3 & 0 \\ -1 & -1 & 1 \\ 1 & 0 & -1 \end{pmatrix} \), \( W = \text{col}(A) \), and \( x = \begin{pmatrix} 3 \\ -4 \\ -1 \\ 2 \end{pmatrix} \). (a) Find \( \text{Proj}_W(x) \). (b) Construct bases for \( W \) and \( \text{Null}(A^\top) \) and verify that every vector in a basis for \( W \) is orthogonal to every vector in a basis in \( \text{Null}(A^\top) \). (c) Use the Gram-Schmidt process to find an orthogonal basis for \( W \).
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