**Problem Statement:**Find the vector $ \mathbf{w} $ if its coordinate vector relative to the basis $ S $ is given by:\[[w]_S = \begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix}\]The basis $ S $ consists of the following vectors:\[S = \left\{ \begin{bmatrix} 7 \\ 0 \\ -10 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} -2 \\ 0 \\ 3 \end{bmatrix} \right\}\]Determine the vector $ \mathbf{w} $ in standard coordinates.**Solution:**To find $ \mathbf{w} $, we use the following expression, given that $ w = a\mathbf{v_1} + b\mathbf{v_2} + c\mathbf{v_3} $, where $ [w]_S = \begin{bmatrix} a \\ b \\ c \end{bmatrix} $:\[\mathbf{w} = 1 \cdot \begin{bmatrix} 7 \\ 0 \\ -10 \end{bmatrix} + 0 \cdot \begin{bmatrix} 0 \\ 1 \\ 3 \end{bmatrix} + 3 \cdot \begin{bmatrix} -2 \\ 0 \\ 3 \end{bmatrix}\]Calculate each part:\[= \begin{bmatrix} 7 \\ 0 \\ -10 \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} + \begin{bmatrix} -6 \\ 0 \\ 9 \end{bmatrix}\]Adding these vectors gives:\[\mathbf{w} = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}\]Thus, the vector $ \mathbf{w} $ in standard coordinates is:\[\mathbf{w} = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}\]

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Answered: 7 Find w if (w)s = |0| relative to the…

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7 Find w if (w)s = |0| relative to the basis S = 3 – 10 3 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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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:**

Find the vector $ \mathbf{w} $ if its coordinate vector relative to the basis $ S $ is given by:

\[
[w]_S = \begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix}
\]

The basis $ S $ consists of the following vectors:

\[
S = \left\{ \begin{bmatrix} 7 \\ 0 \\ -10 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} -2 \\ 0 \\ 3 \end{bmatrix} \right\}
\]

Determine the vector $ \mathbf{w} $ in standard coordinates.

**Solution:**

To find $ \mathbf{w} $, we use the following expression, given that $ w = a\mathbf{v_1} + b\mathbf{v_2} + c\mathbf{v_3} $, where $ [w]_S = \begin{bmatrix} a \\ b \\ c \end{bmatrix} $:

\[
\mathbf{w} = 1 \cdot \begin{bmatrix} 7 \\ 0 \\ -10 \end{bmatrix} + 0 \cdot \begin{bmatrix} 0 \\ 1 \\ 3 \end{bmatrix} + 3 \cdot \begin{bmatrix} -2 \\ 0 \\ 3 \end{bmatrix}
\]

Calculate each part:

\[
= \begin{bmatrix} 7 \\ 0 \\ -10 \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} + \begin{bmatrix} -6 \\ 0 \\ 9 \end{bmatrix}
\]

Adding these vectors gives:

\[
\mathbf{w} = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}
\]

Thus, the vector $ \mathbf{w} $ in standard coordinates is:

\[
\mathbf{w} = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}
\]

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