Using Matrix Reduced Row Echelon Form (RREF), Determine Which Set Of Vectors In { B={\vec{v}_1, \vec{u}_2} $}$ Forms A Basis Of { \mathbb{R}^2$}$.1. { \vec{v}_1=(2,8), \vec{v}_2=(2,5)$} 2. \[ 2. \[ 2. \[ \vec{w}_1=(1,3),

Mar 9, 2025 by ADMIN 221 views

Determining Basis of Vector Spaces using Matrix Reduced Row Echelon Form (RREF)

In linear algebra, a basis of a vector space is a set of vectors that spans the space and is linearly independent. Determining whether a set of vectors forms a basis of a vector space is crucial in various applications, including computer graphics, data analysis, and machine learning. In this article, we will explore how to use Matrix Reduced Row Echelon Form (RREF) to determine which set of vectors forms a basis of ${\mathbb{R}^2}$ .

What is Matrix Reduced Row Echelon Form (RREF)?

Matrix Reduced Row Echelon Form (RREF) is a method of transforming a matrix into a simplified form using elementary row operations. The resulting matrix has the following properties:

All rows consisting entirely of zeros are grouped at the bottom of the matrix.
Each row that is not entirely zeros has a 1 as its first nonzero entry (this entry is called a pivot).
The column in which a pivot is found has all zeros elsewhere, so a column containing a pivot will have zeros everywhere except for one place.

Step 1: Forming the Matrix

To determine whether a set of vectors forms a basis of ${\mathbb{R}^2}$ , we need to form a matrix with the vectors as its columns. Let's consider the two sets of vectors:

${\vec{v}_1=(2,8), \vec{v}_2=(2,5)}$
${\vec{w}_1=(1,3), \vec{w}_2=(4,6)}$

We will form two matrices, one for each set of vectors:

Matrix 1

| 2  8 |
| 2  5 |

Matrix 2

| 1  3 |
| 4  6 |

Step 2: Converting to RREF

To convert a matrix to RREF, we need to perform elementary row operations. Let's start with Matrix 1:

Matrix 1 (RREF)

| 1  4 |
| 0  1 |

We obtained this matrix by performing the following row operations:

Multiply row 1 by 1/2 to get a pivot of 1.
Subtract 2 times row 1 from row 2 to get a zero in the first column.

Now, let's convert Matrix 2 to RREF:

Matrix 2 (RREF)

| 1  1 |
| 0  1 |

We obtained this matrix by performing the following row operations:

Multiply row 1 by 1/4 to get a pivot of 1.
Subtract 4 times row 1 from row 2 to get a zero in the first column.

Step 3: Determining Linear Independence

A set of vectors is linearly independent if and only if the matrix formed by the vectors as columns has a pivot in every column. Let's examine the RREF matrices:

Matrix 1 (RREF) has a pivot in every column, so the vectors ${\vec{v}_1}$ and ${\vec{v}_2}$ are linearly independent.
Matrix 2 (RREF) has a pivot in every column, so the vectors ${\vec{w}_1}$ and ${\vec{w}_2}$ are linearly independent.

In conclusion, we have used Matrix Reduced Row Echelon Form (RREF) to determine which set of vectors forms a basis of ${\mathbb{R}^2}$ . We formed two matrices, one for each set of vectors, and converted them to RREF. We then examined the RREF matrices to determine whether the vectors are linearly independent. The vectors ${\vec{v}_1}$ and ${\vec{v}_2}$ are linearly independent, so they form a basis of ${\mathbb{R}^2}$ . The vectors ${\vec{w}_1}$ and ${\vec{w}_2}$ are also linearly independent, so they form a basis of ${\mathbb{R}^2}$ .

The final answer is that both sets of vectors, ${\{\vec{v}_1, \vec{v}_2\}}$ and ${\{\vec{w}_1, \vec{w}_2\}}$ , form a basis of ${\mathbb{R}^2}$ .
Q&A: Determining Basis of Vector Spaces using Matrix Reduced Row Echelon Form (RREF)

Q: What is the purpose of using Matrix Reduced Row Echelon Form (RREF) in determining basis of vector spaces?

A: The purpose of using Matrix Reduced Row Echelon Form (RREF) is to simplify a matrix and determine whether the vectors in the matrix are linearly independent. This is crucial in determining whether a set of vectors forms a basis of a vector space.

Q: How do I form a matrix from a set of vectors?

A: To form a matrix from a set of vectors, you need to place the vectors as columns in the matrix. For example, if you have two vectors ${\vec{v}_1=(2,8)}$ and ${\vec{v}_2=(2,5)}$ , you would form the following matrix:

| 2 8 | | 2 5 |

Q: What are the properties of a Matrix Reduced Row Echelon Form (RREF)?

A: A Matrix Reduced Row Echelon Form (RREF) has the following properties:

All rows consisting entirely of zeros are grouped at the bottom of the matrix.
Each row that is not entirely zeros has a 1 as its first nonzero entry (this entry is called a pivot).
The column in which a pivot is found has all zeros elsewhere, so a column containing a pivot will have zeros everywhere except for one place.

Q: How do I convert a matrix to RREF?

A: To convert a matrix to RREF, you need to perform elementary row operations. These operations include:

Multiplying a row by a non-zero scalar
Adding a multiple of one row to another row
Interchanging two rows

Q: What does it mean if a matrix has a pivot in every column?

A: If a matrix has a pivot in every column, it means that the vectors in the matrix are linearly independent. This is a necessary and sufficient condition for a set of vectors to form a basis of a vector space.

Q: Can a set of vectors be linearly independent and not form a basis of a vector space?

A: No, a set of vectors cannot be linearly independent and not form a basis of a vector space. If a set of vectors is linearly independent, it means that they span the entire vector space, and therefore form a basis.

Q: How do I determine whether a set of vectors forms a basis of a vector space?

A: To determine whether a set of vectors forms a basis of a vector space, you need to form a matrix with the vectors as columns and convert it to RREF. If the matrix has a pivot in every column, the vectors are linearly independent and form a basis of the vector space.

Q: What are some common mistakes to avoid when using Matrix Reduced Row Echelon Form (RREF)?

A: Some common mistakes to avoid when using Matrix Reduced Row Echelon Form (RREF) include:

Not performing enough row operations to achieve RREF
Performing row operations incorrectly
Not checking for linear independence after converting the matrix to RREF

Q: Can Matrix Reduced Row Echelon Form (RREF) be used to solve systems of linear equations?

A: Yes, Matrix Reduced Row Echelon Form (RREF) can be used to solve systems of linear equations. By converting the coefficient matrix to RREF, you can easily determine the solution to the system of equations.

Q: What are some real-world applications of Matrix Reduced Row Echelon Form (RREF)?

A: Some real-world applications of Matrix Reduced Row Echelon Form (RREF) include:

Computer graphics: Matrix RREF is used to perform transformations on 2D and 3D objects.
Data analysis: Matrix RREF is used to analyze and visualize large datasets.
Machine learning: Matrix RREF is used to train and test machine learning models.