### Finding a Basis for the Subspace of $ \mathbb{R}^3 $ Spanned by $ S $Given:\[ S = \{ (5, 5, 9), (1, 1, 2), (1, 1, 1) \} \]#### **Step 1: Find the Reduced Row-Echelon Form (RREF)**To find the basis for the subspace of $ \mathbb{R}^3 $ spanned by $ S $, we need to find the reduced row-echelon form of the matrix formed by the vectors in $ S $.The matrix whose rows are the vectors in $ S $ is:\[\begin{pmatrix}5 & 5 & 9 \\1 & 1 & 2 \\1 & 1 & 1 \end{pmatrix}\]##### Explanation of Graphical Elements:- **Matrix Setup:** The image contains a visual representation of a matrix setup. Each cell represents an element of the matrix, and it appears that there are arrows and outlines indicating manipulation of rows during the process of finding the RREF. - **Empty Cells:** These boxes are placeholders for the elements of the matrix. - **Arrows:** - The green arrows pointing to the right and downward indicate the direction of some row operations to transform the matrix. - There is a blue dashed outline around one cell, signifying the conversion or row operation centered around that pivot element.In summary, proceed to use standard row operations (swap rows, multiply rows by a scalar, and add/subtract rows) to transform this matrix into its reduced row-echelon form.

In the Row Reduced Echelon form of a matrix, the pivot element is 1 and all elements below the pivot…

Answered: STEP 1: Find the reduced row-echelon…

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STEP 1: Find the reduced row-echelon form of the matrix whose rows are the vectors in S. 00

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

### Finding a Basis for the Subspace of $ \mathbb{R}^3 $ Spanned by $ S $

Given:
\[ S = \{ (5, 5, 9), (1, 1, 2), (1, 1, 1) \} \]

#### **Step 1: Find the Reduced Row-Echelon Form (RREF)**

To find the basis for the subspace of $ \mathbb{R}^3 $ spanned by $ S $, we need to find the reduced row-echelon form of the matrix formed by the vectors in $ S $.

The matrix whose rows are the vectors in $ S $ is:

\[
\begin{pmatrix}
5 & 5 & 9 \\
1 & 1 & 2 \\
1 & 1 & 1
\end{pmatrix}
\]

##### Explanation of Graphical Elements:

- **Matrix Setup:** The image contains a visual representation of a matrix setup. Each cell represents an element of the matrix, and it appears that there are arrows and outlines indicating manipulation of rows during the process of finding the RREF.
- **Empty Cells:** These boxes are placeholders for the elements of the matrix.
- **Arrows:**
- The green arrows pointing to the right and downward indicate the direction of some row operations to transform the matrix.
- There is a blue dashed outline around one cell, signifying the conversion or row operation centered around that pivot element.

In summary, proceed to use standard row operations (swap rows, multiply rows by a scalar, and add/subtract rows) to transform this matrix into its reduced row-echelon form.

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