2. Problem 2 Let A = 2 1 -1 1 -3 -1 1 0 1 1 " W = col(A), and x = 3 -4 -1 2 (a) Find Projw(x). (b) Construct bases for W and Null(AT) and verify that every vector in basis for W is orthogonal to every vector in a basis in Null(AT). (c) Use the Gram-Schmidt process to find an orthogonal basis for W.
2. Problem 2 Let A = 2 1 -1 1 -3 -1 1 0 1 1 " W = col(A), and x = 3 -4 -1 2 (a) Find Projw(x). (b) Construct bases for W and Null(AT) and verify that every vector in basis for W is orthogonal to every vector in a basis in Null(AT). (c) Use the Gram-Schmidt process to find an orthogonal basis for W.
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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Question
answer for parts b and c
![**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 \( \text{Null}(A^T) \) and verify that every vector in the basis for \( W \) is orthogonal to every vector in a basis in \( \text{Null}(A^T) \).
(c) Use the Gram-Schmidt process to find an orthogonal basis for \( W \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd0f3171b-001d-4e71-b013-5e26782f20a2%2Fe2199980-d3c6-4461-ae0a-d3e98c11f3fd%2F7ttk4kg_processed.png&w=3840&q=75)
Transcribed Image Text:**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 \( \text{Null}(A^T) \) and verify that every vector in the basis for \( W \) is orthogonal to every vector in a basis in \( \text{Null}(A^T) \).
(c) Use the Gram-Schmidt process to find an orthogonal basis for \( W \).
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