Which Set Of Vectors In { B=\left{\vec{u}_1, \vec{u}_2\right} $}$ Forms A Basis Of { \mathbb{R}^2 $}$?1. { \vec{v}_1=(2,8), \vec{v}_2=(2,5)$}$2. { \vec{v}_1=(1,3), \vec{v}_2=(2,6)$} 3. \[ 3. \[ 3. \[ \vec{u}_1=(1,4),

Which Set of Vectors Forms a Basis of { \mathbb{R}^2 $}$?

A basis of a vector space is a set of vectors that spans the space and is linearly independent. In the context of { \mathbb{R}^2 $}$, a basis is a set of two linearly independent vectors that span the entire plane. In this article, we will examine three sets of vectors and determine which one forms a basis of { \mathbb{R}^2 $}$.

Understanding Linear Independence

Before we begin, let's recall the definition of linear independence. A set of vectors {\left{\vec{v}_1, \vec{v}_2\right} $}$ is said to be linearly independent if the equation {a\vec{v}_1 + b\vec{v}_2 = \vec{0} $}$ implies that {a = b = 0 $}$. In other words, the only way to express the zero vector as a linear combination of the vectors is with coefficients of zero.

Set 1: {\vec{v}_1=(2,8), \vec{v}_2=(2,5)$}$

To determine if this set of vectors forms a basis, we need to check if they are linearly independent. Let's examine the equation {a\vec{v}_1 + b\vec{v}_2 = \vec{0} $}$. Substituting the values of {\vec{v}_1 $}$ and {\vec{v}_2 $}$, we get:

{a(2,8) + b(2,5) = (0,0) $}$

This equation can be broken down into two separate equations:

${2a + 2b = 0 \$} ${8a + 5b = 0 \$}

We can solve this system of equations using substitution or elimination. Let's use elimination. Multiplying the first equation by 5 and the second equation by 2, we get:

${10a + 10b = 0 \$} ${16a + 10b = 0 \$}

Subtracting the first equation from the second, we get:

${6a = 0 \$}

This implies that {a = 0 $}$. Substituting this value back into the first equation, we get:

${2(0) + 2b = 0 \$}

This implies that {b = 0 $}$. Therefore, the only solution to the equation {a\vec{v}_1 + b\vec{v}_2 = \vec{0} $}$ is {a = b = 0 $}$, which means that the vectors {\vec{v}_1 $}$ and {\vec{v}_2 $}$ are linearly independent.

However, we also need to check if these vectors span { \mathbb{R}^2 $}$. A set of two vectors spans { \mathbb{R}^2 $}$ if and only if the vectors are linearly independent. Since we have already established that the vectors are linearly independent, we can conclude that they also span { \mathbb{R}^2 $}$.

Therefore, the set of vectors {\vec{v}_1=(2,8), \vec{v}_2=(2,5)$}$ forms a basis of { \mathbb{R}^2 $}$.

Set 2: {\vec{v}_1=(1,3), \vec{v}_2=(2,6)$}$

To determine if this set of vectors forms a basis, we need to check if they are linearly independent. Let's examine the equation {a\vec{v}_1 + b\vec{v}_2 = \vec{0} $}$. Substituting the values of {\vec{v}_1 $}$ and {\vec{v}_2 $}$, we get:

{a(1,3) + b(2,6) = (0,0) $}$

This equation can be broken down into two separate equations:

{a + 2b = 0 $}$${3a + 6b = 0 \$}

We can solve this system of equations using substitution or elimination. Let's use elimination. Multiplying the first equation by 3 and the second equation by 1, we get:

${3a + 6b = 0 \$} ${3a + 6b = 0 \$}

Subtracting the first equation from the second, we get:

${0 = 0 \$}

This implies that the system of equations has infinitely many solutions. Therefore, the vectors {\vec{v}_1 $}$ and {\vec{v}_2 $}$ are linearly dependent.

Since the vectors are linearly dependent, they do not span { \mathbb{R}^2 $}$. Therefore, the set of vectors {\vec{v}_1=(1,3), \vec{v}_2=(2,6)$}$ does not form a basis of { \mathbb{R}^2 $}$.

Set 3: {\vec{u}_1=(1,4), \vec{u}_2=(2,3)$}$

To determine if this set of vectors forms a basis, we need to check if they are linearly independent. Let's examine the equation {a\vec{u}_1 + b\vec{u}_2 = \vec{0} $}$. Substituting the values of {\vec{u}_1 $}$ and {\vec{u}_2 $}$, we get:

{a(1,4) + b(2,3) = (0,0) $}$

This equation can be broken down into two separate equations:

{a + 2b = 0 $}$${4a + 3b = 0 \$}

We can solve this system of equations using substitution or elimination. Let's use elimination. Multiplying the first equation by 3 and the second equation by 1, we get:

${3a + 6b = 0 \$} ${4a + 3b = 0 \$}

Subtracting the first equation from the second, we get:

{a - 3b = 0 $}$

This implies that {a = 3b $}$. Substituting this value back into the first equation, we get:

${3b + 2b = 0 \$}

This implies that ${5b = 0 \$}. Therefore, {b = 0 $}$. Substituting this value back into the equation {a = 3b $}$, we get:

{a = 3(0) $}$

This implies that {a = 0 $}$. Therefore, the only solution to the equation {a\vec{u}_1 + b\vec{u}_2 = \vec{0} $}$ is {a = b = 0 $}$, which means that the vectors {\vec{u}_1 $}$ and {\vec{u}_2 $}$ are linearly independent.

However, we also need to check if these vectors span { \mathbb{R}^2 $}$. A set of two vectors spans { \mathbb{R}^2 $}$ if and only if the vectors are linearly independent. Since we have already established that the vectors are linearly independent, we can conclude that they also span { \mathbb{R}^2 $}$.

Therefore, the set of vectors {\vec{u}_1=(1,4), \vec{u}_2=(2,3)$}$ forms a basis of { \mathbb{R}^2 $}$.

Conclusion

In this article, we examined three sets of vectors and determined which one forms a basis of { \mathbb{R}^2 $}$. We found that the set of vectors {\vec{v}_1=(2,8), \vec{v}_2=(2,5)$}$ and the set of vectors {\vec{u}_1=(1,4), \vec{u}_2=(2,3)$}$ both form a basis of { \mathbb{R}^2 $}$. On the other hand, the set of vectors {\vec{v}_1=(1,3), \vec{v}_2=(2,6)$}$ does not form a basis of { \mathbb{R}^2 $}$.

References

• [1] Hoffman, K., & Kunze, R. (1971). Linear Algebra. Prentice Hall.
• [2] Strang, G. (1988). Linear Algebra and Its Applications. Academic Press.

Glossary

• Basis: A set of vectors that spans a vector space and is linearly independent.
• Linear Independence: A set of vectors is said to be linearly independent if the equation {a\vecv}_1 + b\vec{v}_2 = \vec{0} $}$ implies that {a = b
  **Q&A Which Set of Vectors Forms a Basis of [$ \mathbb{R^2 $}$?**

In our previous article, we examined three sets of vectors and determined which one forms a basis of { \mathbb{R}^2 $}$. We found that the set of vectors {\vec{v}_1=(2,8), \vec{v}_2=(2,5)$}$ and the set of vectors {\vec{u}_1=(1,4), \vec{u}_2=(2,3)$}$ both form a basis of { \mathbb{R}^2 $}$. In this article, we will answer some frequently asked questions about the topic.

Q: What is a basis of a vector space?

A: A basis of a vector space is a set of vectors that spans the space and is linearly independent. In other words, a basis is a set of vectors that can be used to express any vector in the space as a linear combination of the basis vectors.

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

A: To determine if a set of vectors forms a basis of a vector space, you need to check if the vectors are linearly independent and span the space. You can do this by examining the equation {a\vec{v}_1 + b\vec{v}_2 = \vec{0} $}$ and checking if the only solution is {a = b = 0 $}$.

Q: What is linear independence?

A: Linear independence is a property of a set of vectors that means the only way to express the zero vector as a linear combination of the vectors is with coefficients of zero. In other words, a set of vectors is linearly independent if the equation {a\vec{v}_1 + b\vec{v}_2 = \vec{0} $}$ implies that {a = b = 0 $}$.

Q: How do I check if a set of vectors is linearly independent?

A: To check if a set of vectors is linearly independent, you need to examine the equation {a\vec{v}_1 + b\vec{v}_2 = \vec{0} $}$ and check if the only solution is {a = b = 0 $}$. You can do this by solving the system of equations using substitution or elimination.

Q: What is the difference between a basis and a spanning set?

A: A basis is a set of vectors that spans a vector space and is linearly independent. A spanning set is a set of vectors that spans a vector space, but may not be linearly independent. In other words, a basis is a set of vectors that can be used to express any vector in the space as a linear combination of the basis vectors, and the vectors are also linearly independent.

Q: Can a set of vectors be both a basis and a spanning set?

A: Yes, a set of vectors can be both a basis and a spanning set. In fact, a basis is a set of vectors that is both a spanning set and linearly independent.

Q: How do I find a basis for a vector space?

A: To find a basis for a vector space, you need to find a set of vectors that spans the space and is linearly independent. You can do this by examining the equation {a\vec{v}_1 + b\vec{v}_2 = \vec{0} $}$ and checking if the only solution is {a = b = 0 $}$. You can also use the Gram-Schmidt process to find a basis for a vector space.

Q: What is the Gram-Schmidt process?

A: The Gram-Schmidt process is a method for finding a basis for a vector space. It involves taking a set of vectors and applying a series of transformations to produce a new set of vectors that is linearly independent and spans the space.

Q: Can I use the Gram-Schmidt process to find a basis for any vector space?

A: Yes, you can use the Gram-Schmidt process to find a basis for any vector space. However, the process may not always produce a basis that is easy to work with, and it may require a lot of computation.

Q: What are some common mistakes to avoid when finding a basis for a vector space?

A: Some common mistakes to avoid when finding a basis for a vector space include:

• Not checking if the vectors are linearly independent
• Not checking if the vectors span the space
• Using a set of vectors that is not linearly independent
• Using a set of vectors that does not span the space

Q: How do I know if a set of vectors is a basis for a vector space?

A: To know if a set of vectors is a basis for a vector space, you need to check if the vectors are linearly independent and span the space. You can do this by examining the equation {a\vec{v}_1 + b\vec{v}_2 = \vec{0} $}$ and checking if the only solution is {a = b = 0 $}$.

Q: What are some real-world applications of bases and spanning sets?

A: Bases and spanning sets have many real-world applications, including:

• Data analysis and machine learning
• Signal processing and image analysis
• Computer graphics and game development
• Physics and engineering

Q: Can I use a basis to solve a system of linear equations?

A: Yes, you can use a basis to solve a system of linear equations. In fact, a basis is a set of vectors that can be used to express any vector in the space as a linear combination of the basis vectors, and the vectors are also linearly independent.

Q: How do I use a basis to solve a system of linear equations?

A: To use a basis to solve a system of linear equations, you need to express the system of equations in terms of the basis vectors. You can do this by examining the equation {a\vec{v}_1 + b\vec{v}_2 = \vec{0} $}$ and checking if the only solution is {a = b = 0 $}$. You can then use the basis vectors to solve the system of equations.

Q: What are some common mistakes to avoid when using a basis to solve a system of linear equations?

A: Some common mistakes to avoid when using a basis to solve a system of linear equations include:

• Not checking if the vectors are linearly independent
• Not checking if the vectors span the space
• Using a set of vectors that is not linearly independent
• Using a set of vectors that does not span the space

Q: How do I know if a set of vectors is a basis for a vector space?

A: To know if a set of vectors is a basis for a vector space, you need to check if the vectors are linearly independent and span the space. You can do this by examining the equation {a\vec{v}_1 + b\vec{v}_2 = \vec{0} $}$ and checking if the only solution is {a = b = 0 $}$.

Q: What are some real-world applications of bases and spanning sets in data analysis and machine learning?

A: Bases and spanning sets have many real-world applications in data analysis and machine learning, including:

• Dimensionality reduction
• Feature extraction
• Clustering and classification
• Regression and prediction

Q: Can I use a basis to solve a system of linear equations in data analysis and machine learning?

A: Yes, you can use a basis to solve a system of linear equations in data analysis and machine learning. In fact, a basis is a set of vectors that can be used to express any vector in the space as a linear combination of the basis vectors, and the vectors are also linearly independent.

Q: How do I use a basis to solve a system of linear equations in data analysis and machine learning?

A: To use a basis to solve a system of linear equations in data analysis and machine learning, you need to express the system of equations in terms of the basis vectors. You can do this by examining the equation {a\vec{v}_1 + b\vec{v}_2 = \vec{0} $}$ and checking if the only solution is {a = b = 0 $}$. You can then use the basis vectors to solve the system of equations.

Q: What are some common mistakes to avoid when using a basis to solve a system of linear equations in data analysis and machine learning?

A: Some common mistakes to avoid when using a basis to solve a system of linear equations in data analysis and machine learning include:

• Not checking if the vectors are linearly independent
• Not checking if the vectors span the space
• Using a set of vectors that is not linearly independent
• Using a set of vectors that does not span the space