1. Find the coordinates of vector w (1,2, 3) in the basis A which consists of T1 = (1, 1, 0), 72 = (0, –1, 0), ī3 = (1,0, 1). = (0, –1, 0), v3 = (1,0, 1).
1. Find the coordinates of vector w (1,2, 3) in the basis A which consists of T1 = (1, 1, 0), 72 = (0, –1, 0), ī3 = (1,0, 1). = (0, –1, 0), v3 = (1,0, 1).
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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![**Problem Statement**:
1. Find the coordinates of vector \(\vec{w} = \langle 1, 2, 3 \rangle\) in the basis \(A\) which consists of \(\vec{v}_1 = \langle 1, 1, 0 \rangle\), \(\vec{v}_2 = \langle 0, -1, 0 \rangle\), \(\vec{v}_3 = \langle 1, 0, 1 \rangle\).
---
**Explanation**:
The task is to express the vector \(\vec{w}\) in terms of the given basis vectors \(\vec{v}_1\), \(\vec{v}_2\), and \(\vec{v}_3\). This involves solving the system of linear equations to determine the coordinates \(c_1\), \(c_2\), and \(c_3\) such that:
\[
\vec{w} = c_1 \vec{v}_1 + c_2 \vec{v}_2 + c_3 \vec{v}_3
\]
Given:
\[
\vec{w} = \langle 1, 2, 3 \rangle
\]
\[
\vec{v}_1 = \langle 1, 1, 0 \rangle
\]
\[
\vec{v}_2 = \langle 0, -1, 0 \rangle
\]
\[
\vec{v}_3 = \langle 1, 0, 1 \rangle
\]
We need to find the scalars \(c_1\), \(c_2\), and \(c_3\) such that:
\[
\langle 1, 2, 3 \rangle = c_1 \langle 1, 1, 0 \rangle + c_2 \langle 0, -1, 0 \rangle + c_3 \langle 1, 0, 1 \rangle
\]
This is a standard problem in linear algebra that can be solved in various ways, including row reduction, matrix inversion, or using computational tools.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff216d087-3051-47b6-8a65-d5d792badeb4%2F5b86b681-7097-4294-8c84-7ed08934abb8%2Fxcw96_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**:
1. Find the coordinates of vector \(\vec{w} = \langle 1, 2, 3 \rangle\) in the basis \(A\) which consists of \(\vec{v}_1 = \langle 1, 1, 0 \rangle\), \(\vec{v}_2 = \langle 0, -1, 0 \rangle\), \(\vec{v}_3 = \langle 1, 0, 1 \rangle\).
---
**Explanation**:
The task is to express the vector \(\vec{w}\) in terms of the given basis vectors \(\vec{v}_1\), \(\vec{v}_2\), and \(\vec{v}_3\). This involves solving the system of linear equations to determine the coordinates \(c_1\), \(c_2\), and \(c_3\) such that:
\[
\vec{w} = c_1 \vec{v}_1 + c_2 \vec{v}_2 + c_3 \vec{v}_3
\]
Given:
\[
\vec{w} = \langle 1, 2, 3 \rangle
\]
\[
\vec{v}_1 = \langle 1, 1, 0 \rangle
\]
\[
\vec{v}_2 = \langle 0, -1, 0 \rangle
\]
\[
\vec{v}_3 = \langle 1, 0, 1 \rangle
\]
We need to find the scalars \(c_1\), \(c_2\), and \(c_3\) such that:
\[
\langle 1, 2, 3 \rangle = c_1 \langle 1, 1, 0 \rangle + c_2 \langle 0, -1, 0 \rangle + c_3 \langle 1, 0, 1 \rangle
\]
This is a standard problem in linear algebra that can be solved in various ways, including row reduction, matrix inversion, or using computational tools.
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