**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.

Answered: Image /qna-images/answer/5b86b681-7097-4294-8c84-7ed08934abb8.jpg

Answered: 1. Find the coordinates of vector w…

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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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

**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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