Which Two Vectors Are Linearly Independent?A. A ⃗ = ( 1 , 1 \vec{a}=(1,1 A = ( 1 , 1 ] And B ⃗ = ( 4 , 3 \vec{b}=(4,3 B = ( 4 , 3 ]B. A ⃗ = ( 1 , 3 \vec{a}=(1,3 A = ( 1 , 3 ] And B ⃗ = ( 2 , 6 \vec{b}=(2,6 B = ( 2 , 6 ]C. A ⃗ = ( 2 , 3 \vec{a}=(2,3 A = ( 2 , 3 ] And B ⃗ = ( 0 , 0 \vec{b}=(0,0 B = ( 0 , 0 ]

Introduction

In the realm of linear algebra, vectors are fundamental objects used to represent quantities with both magnitude and direction. When dealing with vectors, it's essential to understand the concept of linear independence, which plays a crucial role in various mathematical and scientific applications. In this article, we will delve into the concept of linear independence and explore which two vectors are linearly independent among the given options.

What is Linear Independence?

Two vectors $\vec{a}$ and $\vec{b}$ are said to be linearly independent if the only solution to the equation $c_1\vec{a} + c_2\vec{b} = \vec{0}$ is when $c_1 = c_2 = 0$. In other words, the only way to express the zero vector as a linear combination of $\vec{a}$ and $\vec{b}$ is by using coefficients of zero. This means that neither vector can be expressed as a scalar multiple of the other.

Mathematical Representation

To determine whether two vectors are linearly independent, we can use the following mathematical representation:

$$c_1\vec{a} + c_2\vec{b} = \vec{0}$$

where $\vec{a}$ and $\vec{b}$ are the vectors in question, and $c_1$ and $c_2$ are scalars.

Option A: $\vec{a}=(1,1)$ and $\vec{b}=(4,3)$

Let's examine the first option, where $\vec{a}=(1,1)$ and $\vec{b}=(4,3)$. To determine whether these vectors are linearly independent, we can set up the equation:

$$c_1(1,1) + c_2(4,3) = (0,0)$$

This equation can be rewritten as a system of linear equations:

$$c_1 + 4c_2 = 0$$

$$c_1 + 3c_2 = 0$$

Solving this system of equations, we find that $c_1 = -3c_2$ and $c_2 = -\frac{1}{4}c_1$. Substituting these values into the first equation, we get:

$$-3c_2 + 4c_2 = 0$$

Simplifying this equation, we find that $c_2 = 0$. Substituting this value back into the first equation, we get:

$$-3(0) + 4c_2 = 0$$

This equation is satisfied for any value of $c_2$. Therefore, we can conclude that $c_1 = 0$ as well.

Conclusion for Option A

Based on our analysis, we can conclude that $\vec{a}=(1,1)$ and $\vec{b}=(4,3)$ are linearly dependent. This means that one vector can be expressed as a scalar multiple of the other.

Option B: $\vec{a}=(1,3)$ and $\vec{b}=(2,6)$

Let's examine the second option, where $\vec{a}=(1,3)$ and $\vec{b}=(2,6)$. To determine whether these vectors are linearly independent, we can set up the equation:

$$c_1(1,3) + c_2(2,6) = (0,0)$$

This equation can be rewritten as a system of linear equations:

$$c_1 + 2c_2 = 0$$

$$3c_1 + 6c_2 = 0$$

Solving this system of equations, we find that $c_1 = -2c_2$ and $c_2 = -\frac{1}{3}c_1$. Substituting these values into the first equation, we get:

$$-2c_2 + 2c_2 = 0$$

This equation is satisfied for any value of $c_2$. Therefore, we can conclude that $c_1 = 0$ as well.

Conclusion for Option B

Based on our analysis, we can conclude that $\vec{a}=(1,3)$ and $\vec{b}=(2,6)$ are linearly dependent. This means that one vector can be expressed as a scalar multiple of the other.

Option C: $\vec{a}=(2,3)$ and $\vec{b}=(0,0)$

Let's examine the third option, where $\vec{a}=(2,3)$ and $\vec{b}=(0,0)$. To determine whether these vectors are linearly independent, we can set up the equation:

$$c_1(2,3) + c_2(0,0) = (0,0)$$

This equation can be rewritten as a system of linear equations:

$$2c_1 = 0$$

$$3c_1 = 0$$

Solving this system of equations, we find that $c_1 = 0$.

Conclusion for Option C

Based on our analysis, we can conclude that $\vec{a}=(2,3)$ and $\vec{b}=(0,0)$ are linearly dependent. This means that one vector can be expressed as a scalar multiple of the other.

Conclusion

In conclusion, none of the given options satisfy the condition of linear independence. However, we can see that the vectors in each option are linearly dependent, meaning that one vector can be expressed as a scalar multiple of the other.

Final Answer

Q: What is the difference between linear independence and linear dependence?

A: Linear independence and linear dependence are two fundamental concepts in linear algebra. Linear independence means that a set of vectors cannot be expressed as a linear combination of other vectors, whereas linear dependence means that a set of vectors can be expressed as a linear combination of other vectors.

Q: How do I determine if two vectors are linearly independent?

A: To determine if two vectors are linearly independent, you can set up the equation $c_1\vec{a} + c_2\vec{b} = \vec{0}$ and solve for $c_1$ and $c_2$. If the only solution is $c_1 = c_2 = 0$, then the vectors are linearly independent.

Q: What is the significance of linear independence in linear algebra?

A: Linear independence is a crucial concept in linear algebra because it allows us to determine the dimension of a vector space. A set of vectors is said to be a basis for a vector space if it is linearly independent and spans the entire space. The dimension of a vector space is equal to the number of vectors in a basis.

Q: Can a vector be linearly independent and linearly dependent at the same time?

A: No, a vector cannot be linearly independent and linearly dependent at the same time. If a vector is linearly independent, it means that it cannot be expressed as a linear combination of other vectors. If a vector is linearly dependent, it means that it can be expressed as a linear combination of other vectors.

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

A: To determine if a set of vectors is linearly independent, you can set up the equation $c_1\vec{a} + c_2\vec{b} + \ldots + c_n\vec{n} = \vec{0}$ and solve for $c_1, c_2, \ldots, c_n$. If the only solution is $c_1 = c_2 = \ldots = c_n = 0$, then the set of vectors is linearly independent.

Q: What is the relationship between linear independence and the null space of a matrix?

A: The null space of a matrix is the set of all vectors that satisfy the equation $A\vec{x} = \vec{0}$. If a matrix has a null space with dimension greater than 0, then the columns of the matrix are linearly dependent. Conversely, if the columns of a matrix are linearly independent, then the null space of the matrix has dimension 0.

Q: Can a vector space have multiple bases?

A: Yes, a vector space can have multiple bases. In fact, any set of linearly independent vectors that spans the entire space can be considered a basis for the space.

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

A: To find a basis for a vector space, you can start with a set of linearly independent vectors and add vectors to the set until it spans the entire space. Alternatively, you can use the Gram-Schmidt process to find an orthogonal basis for the space.

Q: What is the significance of the dimension of a vector space?

A: The dimension of a vector space is a fundamental concept in linear algebra because it allows us to determine the number of vectors in a basis for the space. The dimension of a vector space is also equal to the number of linearly independent vectors in the space.

Q: Can a vector space have a dimension of 0?

A: Yes, a vector space can have a dimension of 0. This occurs when the space contains only the zero vector.

Q: What is the relationship between the dimension of a vector space and the number of linearly independent vectors?

A: The dimension of a vector space is equal to the number of linearly independent vectors in the space. This means that if a vector space has a dimension of n, then it contains n linearly independent vectors.