Which Set Of Vectors In { B = \left{\vec{w}_1, \vec{w}_2\right} $}$ Forms A Basis For { \mathbb{R}^2$}$?A. { \vec{u}_1 = (2, 8), \vec{u}_2 = (2, 5)$}$B. { \vec{w}_1 = (1, 3), \vec{w}_2 = (2, 6)$}$C.
Which Set of Vectors Forms a Basis for {\mathbb{R}^2$}$?
A basis for 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 would be a set of two vectors that can be used to express any vector in {\mathbb{R}^2$}$ as a linear combination of the two vectors.
Understanding the Concept of Basis
To determine which set of vectors forms a basis for {\mathbb{R}^2$}$, we need to understand the concept of basis and how it applies to vector spaces. A basis for a vector space is a set of vectors that satisfies two properties:
- Span: The set of vectors must span the vector space, meaning that any vector in the space can be expressed as a linear combination of the vectors in the set.
- Linear Independence: The set of vectors must be linearly independent, meaning that none of the vectors in the set can be expressed as a linear combination of the other vectors in the set.
Analyzing the Options
We are given three sets of vectors to consider:
A. {\vec{u}_1 = (2, 8), \vec{u}_2 = (2, 5)$}$ B. {\vec{w}_1 = (1, 3), \vec{w}_2 = (2, 6)$}$ C. {\vec{v}_1 = (1, 0), \vec{v}_2 = (0, 1)$}$
To determine which set of vectors forms a basis for {\mathbb{R}^2$}$, we need to analyze each set and determine whether it satisfies the two properties of a basis.
Option A: {\vec{u}_1 = (2, 8), \vec{u}_2 = (2, 5)$}$
To determine whether this set of vectors forms a basis for {\mathbb{R}^2$}$, we need to check whether it spans the space and is linearly independent.
Span
To check whether this set of vectors spans the space, we need to determine whether any vector in {\mathbb{R}^2$}$ can be expressed as a linear combination of {\vec{u}_1$}$ and {\vec{u}_2$}$. We can do this by solving the equation:
{a\vec{u}_1 + b\vec{u}_2 = \vec{x}$}$
where {\vec{x}$}$ is any vector in {\mathbb{R}^2$}$.
Substituting the values of {\vec{u}_1$}$ and {\vec{u}_2$}$, we get:
{a(2, 8) + b(2, 5) = (x_1, x_2)$}$
This equation can be rewritten as a system of linear equations:
${2a + 2b = x_1\$} ${8a + 5b = x_2\$}
We can solve this system of equations using substitution or elimination. Let's use substitution.
Rearranging the first equation, we get:
{a = \frac{x_1 - 2b}{2}$}$
Substituting this expression for {a$}$ into the second equation, we get:
${8\left(\frac{x_1 - 2b}{2}\right) + 5b = x_2\$}
Simplifying this equation, we get:
${4x_1 - 16b + 5b = 2x_2\$}
Combine like terms:
${4x_1 - 11b = 2x_2\$}
Now, we can see that this equation is not true for all values of {x_1$}$ and {x_2$}$. Therefore, this set of vectors does not span the space.
Linear Independence
Since this set of vectors does not span the space, we do not need to check whether it is linearly independent.
Option B: {\vec{w}_1 = (1, 3), \vec{w}_2 = (2, 6)$}$
To determine whether this set of vectors forms a basis for {\mathbb{R}^2$}$, we need to check whether it spans the space and is linearly independent.
Span
To check whether this set of vectors spans the space, we need to determine whether any vector in {\mathbb{R}^2$}$ can be expressed as a linear combination of {\vec{w}_1$}$ and {\vec{w}_2$}$. We can do this by solving the equation:
{a\vec{w}_1 + b\vec{w}_2 = \vec{x}$}$
where {\vec{x}$}$ is any vector in {\mathbb{R}^2$}$.
Substituting the values of {\vec{w}_1$}$ and {\vec{w}_2$}$, we get:
{a(1, 3) + b(2, 6) = (x_1, x_2)$}$
This equation can be rewritten as a system of linear equations:
{a + 2b = x_1$}{3a + 6b = x_2\$}
We can solve this system of equations using substitution or elimination. Let's use substitution.
Rearranging the first equation, we get:
{a = x_1 - 2b$}$
Substituting this expression for {a$}$ into the second equation, we get:
${3(x_1 - 2b) + 6b = x_2\$}
Simplifying this equation, we get:
${3x_1 - 6b + 6b = x_2\$}
Combine like terms:
${3x_1 = x_2\$}
Now, we can see that this equation is true for all values of {x_1$}$ and {x_2$}$. Therefore, this set of vectors spans the space.
Linear Independence
To check whether this set of vectors is linearly independent, we need to determine whether the equation:
{a\vec{w}_1 + b\vec{w}_2 = \vec{0}$}$
has only the trivial solution {a = b = 0$}$.
Substituting the values of {\vec{w}_1$}$ and {\vec{w}_2$}$, we get:
{a(1, 3) + b(2, 6) = (0, 0)$}$
This equation can be rewritten as a system of linear equations:
{a + 2b = 0$}{3a + 6b = 0\$}
We can solve this system of equations using substitution or elimination. Let's use substitution.
Rearranging the first equation, we get:
{a = -2b$}$
Substituting this expression for {a$}$ into the second equation, we get:
${3(-2b) + 6b = 0\$}
Simplifying this equation, we get:
{-6b + 6b = 0$}$
Combine like terms:
${0 = 0\$}
This equation is true for all values of {b$}$. Therefore, this set of vectors is linearly independent.
Option C: {\vec{v}_1 = (1, 0), \vec{v}_2 = (0, 1)$}$
To determine whether this set of vectors forms a basis for {\mathbb{R}^2$}$, we need to check whether it spans the space and is linearly independent.
Span
To check whether this set of vectors spans the space, we need to determine whether any vector in {\mathbb{R}^2$}$ can be expressed as a linear combination of {\vec{v}_1$}$ and {\vec{v}_2$}$. We can do this by solving the equation:
{a\vec{v}_1 + b\vec{v}_2 = \vec{x}$}$
where {\vec{x}$}$ is any vector in {\mathbb{R}^2$}$.
Substituting the values of {\vec{v}_1$}$ and {\vec{v}_2$}$, we get:
{a(1, 0) + b(0, 1) = (x_1, x_2)$}$
This equation can be rewritten as a system of linear equations:
{a = x_1$}{b = x_2\$}
We can see that this system of equations has a solution for any values of {x_1$}$ and {x_2$}$. Therefore, this set of vectors spans the space.
Linear Independence
To check whether this set of vectors is linearly independent, we need to determine whether the equation:
{a
Q&A: Which Set of Vectors Forms a Basis for [\mathbb{R}^2\$}?
In the previous article, we analyzed three sets of vectors to determine which one forms a basis for {\mathbb{R}^2$}$. We found that only one of the sets satisfies the two properties of a basis: spanning the space and being linearly independent.
Q: What is a basis for a vector space?
A: A basis for 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 vectors in the set.
Q: Why is it important to have a basis for a vector space?
A: Having a basis for a vector space is important because it allows us to express any vector in the space in a unique way. This makes it easier to perform calculations and operations on vectors.
Q: How do I determine whether a set of vectors forms a basis for a vector space?
A: To determine whether a set of vectors forms a basis for a vector space, you need to check whether it spans the space and is linearly independent. You can do this by solving the equation:
{a\vec{v}_1 + b\vec{v}_2 = \vec{x}$}$
where {\vec{x}$}$ is any vector in the space.
Q: What is the difference between a basis and a spanning set?
A: A basis is a set of vectors that spans the space and is linearly independent. A spanning set is a set of vectors that spans the space, but may not be 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, any set of vectors that forms a basis for a vector space is also a spanning set.
Q: How do I find a basis for a vector space?
A: To find a basis for a vector space, you can use the following steps:
- Find a spanning set: Find a set of vectors that spans the space.
- Check for linear independence: Check whether the set of vectors is linearly independent.
- Remove any linearly dependent vectors: If the set of vectors is not linearly independent, remove any vectors that are linearly dependent.
- Check again for linear independence: Check whether the remaining set of vectors is linearly independent.
Q: Can a basis for a vector space be infinite?
A: No, a basis for a vector space cannot be infinite. A basis is a finite set of vectors that spans the space and is linearly independent.
Q: Can a basis for a vector space be empty?
A: No, a basis for a vector space cannot be empty. A basis is a non-empty set of vectors that spans the space and is linearly independent.
Q: What is the significance of a basis for a vector space in real-world applications?
A: A basis for a vector space has many significant applications in real-world problems, such as:
- Linear algebra: A basis for a vector space is used to solve systems of linear equations and to find the inverse of a matrix.
- Computer graphics: A basis for a vector space is used to represent 3D objects and to perform transformations on them.
- Signal processing: A basis for a vector space is used to represent signals and to perform operations on them.
In conclusion, a basis for a vector space is a set of vectors that spans the space and is linearly independent. It is an important concept in linear algebra and has many significant applications in real-world problems.