Describe all solutions of Ax = 0 in vector parametric form, where A is row equivalent to the matrix given below:(see attached image) On this educational page, we will explore the following matrix, which is commonly used in linear algebra and related fields such as computer science, physics, and engineering.The matrix is as follows:\[\begin{bmatrix}1 & 2 & 0 & 0 & 1 & -1 \\0 & 0 & 2 & 0 & -1 & 1 \\0 & 0 & 0 & 1 & 1 & -1 \\\end{bmatrix}\]**Explanation:**This is a $ 3 \times 6 $ matrix, denoted as having 3 rows and 6 columns.**Row 1:** \[ \begin{bmatrix}1 & 2 & 0 & 0 & 1 & -1 \end{bmatrix} \]**Row 2:** \[ \begin{bmatrix}0 & 0 & 2 & 0 & -1 & 1\end{bmatrix} \]**Row 3:** \[ \begin{bmatrix}0 & 0 & 0 & 1 & 1 & -1\end{bmatrix} \]### Key Points:- The elements of each row and column can represent different quantities, depending on the context, such as coefficients in a system of equations, or transformations in vector space.- In this particular matrix: - The first row contains the elements: $1, 2, 0, 0, 1, -1$ - The second row contains the elements: $0, 0, 2, 0, -1, 1$ - The third row contains the elements: $0, 0, 0, 1, 1, -1$### Applications:- **Systems of Equations:** This matrix might be used to represent a system of linear equations where each row represents an equation and each column represents a variable.- **Transformation Matrices:** It could also represent a transformation in a higher-dimensional vector space, where matrix multiplication alters the components of a vector according to the transformation rules.### Conclusion:Understanding how to read and work with matrices like this one is critical in various scientific and engineering disciplines. This basic knowledge is a stepping stone to more complex concepts and applications.For further detailed study, please refer to our sections on 'Matrix Operations'

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Describe all solutions of Ax = 0 in vector parametric form, where A is row equivalent to the matrix given below

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

Describe all solutions of Ax = 0 in vector parametric form, where A is row equivalent to the matrix given below:

(see attached image)

On this educational page, we will explore the following matrix, which is commonly used in linear algebra and related fields such as computer science, physics, and engineering.

The matrix is as follows:

\[
\begin{bmatrix}
1 & 2 & 0 & 0 & 1 & -1 \\
0 & 0 & 2 & 0 & -1 & 1 \\
0 & 0 & 0 & 1 & 1 & -1 \\
\end{bmatrix}
\]

**Explanation:**

This is a $ 3 \times 6 $ matrix, denoted as having 3 rows and 6 columns.

**Row 1:**
\[ \begin{bmatrix}
1 & 2 & 0 & 0 & 1 & -1
\end{bmatrix} \]

**Row 2:**
\[ \begin{bmatrix}
0 & 0 & 2 & 0 & -1 & 1
\end{bmatrix} \]

**Row 3:**
\[ \begin{bmatrix}
0 & 0 & 0 & 1 & 1 & -1
\end{bmatrix} \]

### Key Points:
- The elements of each row and column can represent different quantities, depending on the context, such as coefficients in a system of equations, or transformations in vector space.
- In this particular matrix:
- The first row contains the elements: $1, 2, 0, 0, 1, -1$
- The second row contains the elements: $0, 0, 2, 0, -1, 1$
- The third row contains the elements: $0, 0, 0, 1, 1, -1$

### Applications:

- **Systems of Equations:** This matrix might be used to represent a system of linear equations where each row represents an equation and each column represents a variable.
- **Transformation Matrices:** It could also represent a transformation in a higher-dimensional vector space, where matrix multiplication alters the components of a vector according to the transformation rules.

### Conclusion:

Understanding how to read and work with matrices like this one is critical in various scientific and engineering disciplines. This basic knowledge is a stepping stone to more complex concepts and applications.

For further detailed study, please refer to our sections on 'Matrix Operations'

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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