The Coordinate Vector Of { \left[\begin{array}{l}x \ Y \ Z\end{array}\right]$}$ With Respect To The Basis [$B=\left{\left[\begin{array}{l}1 \ 0 \ 1\end{array}\right],\left[\begin{array}{l}1 \ 1 \

Introduction

In mathematics, particularly in linear algebra, the concept of a coordinate vector is crucial in understanding the relationship between a vector and a basis. A basis is a set of vectors that can be used to express any vector in a vector space as a linear combination of the basis vectors. In this article, we will explore the concept of the coordinate vector of a vector with respect to a basis, and we will use the given example to illustrate the concept.

What is a Coordinate Vector?

A coordinate vector is a vector that represents the coefficients of the linear combination of the basis vectors that express a given vector. In other words, it is a vector that contains the coordinates of the vector with respect to the basis. The coordinate vector is denoted by the symbol $\left[\begin{array}{l}a \\ b \\ c\end{array}\right]$, where $a$, $b$, and $c$ are the coordinates of the vector with respect to the basis.

The Coordinate Vector of a Vector with Respect to a Basis

Given a vector $\left[\begin{array}{l}x \\ y \\ z\end{array}\right]$ and a basis $B=\left\{\left[\begin{array}{l}1 \\ 0 \\ 1\end{array}\right],\left[\begin{array}{l}1 \\ 1 \\ 0\end{array}\right]\right\}$, we want to find the coordinate vector of the vector with respect to the basis. To do this, we need to express the vector as a linear combination of the basis vectors.

Expressing the Vector as a Linear Combination of the Basis Vectors

We can express the vector $\left[\begin{array}{l}x \\ y \\ z\end{array}\right]$ as a linear combination of the basis vectors as follows:

$$\left[\begin{array}{l}x \\ y \\ z\end{array}\right] = a\left[\begin{array}{l}1 \\ 0 \\ 1\end{array}\right] + b\left[\begin{array}{l}1 \\ 1 \\ 0\end{array}\right]$$

where $a$ and $b$ are the coefficients of the linear combination.

Finding the Coefficients of the Linear Combination

To find the coefficients $a$ and $b$, we can equate the corresponding components of the vector and the linear combination:

$$x = a + b$$

$$y = b$$

$$z = a$$

Solving this system of equations, we get:

$$a = \frac{x+z}{2}$$

$$b = \frac{x-y}{2}$$

The Coordinate Vector

The coordinate vector of the vector with respect to the basis is:

$$\left[\begin{array}{l}a \\ b \\ c\end{array}\right] = \left[\begin{array}{l}\frac{x+z}{2} \\ \frac{x-y}{2} \\ \frac{x+z}{2}\end{array}\right]$$

Conclusion

In this article, we have explored the concept of the coordinate vector of a vector with respect to a basis. We have used the given example to illustrate the concept and have shown how to find the coordinate vector of a vector with respect to a basis. The coordinate vector is a crucial concept in linear algebra, and it has many applications in mathematics and science.

Applications of Coordinate Vectors

Coordinate vectors have many applications in mathematics and science. Some of the applications include:

• Linear Transformations: Coordinate vectors are used to represent the coefficients of linear transformations.
• Matrix Representations: Coordinate vectors are used to represent the coefficients of matrix representations of linear transformations.
• Change of Basis: Coordinate vectors are used to represent the coefficients of change of basis transformations.
• Vector Spaces: Coordinate vectors are used to represent the coefficients of vectors in vector spaces.

Conclusion

In conclusion, the coordinate vector of a vector with respect to a basis is a crucial concept in linear algebra. It is used to represent the coefficients of the linear combination of the basis vectors that express a given vector. The coordinate vector has many applications in mathematics and science, and it is an essential tool for understanding linear algebra.

References

• Linear Algebra and Its Applications by Gilbert Strang
• Introduction to Linear Algebra by Jim Hefferon
• Linear Algebra by David C. Lay

Further Reading

For further reading on linear algebra and coordinate vectors, we recommend the following resources:

• Linear Algebra and Its Applications by Gilbert Strang
• Introduction to Linear Algebra by Jim Hefferon
• Linear Algebra by David C. Lay

Glossary

• Basis: A set of vectors that can be used to express any vector in a vector space as a linear combination of the basis vectors.
• Coordinate Vector: A vector that represents the coefficients of the linear combination of the basis vectors that express a given vector.
• Linear Combination: A linear combination of vectors is a sum of scalar multiples of the vectors.
• Vector Space: A set of vectors that is closed under addition and scalar multiplication.
  Frequently Asked Questions (FAQs) about Coordinate Vectors ===========================================================

Q: What is a coordinate vector?

A: A coordinate vector is a vector that represents the coefficients of the linear combination of the basis vectors that express a given vector.

Q: Why are coordinate vectors important?

A: Coordinate vectors are important because they provide a way to represent the coefficients of the linear combination of the basis vectors that express a given vector. This is useful in many areas of mathematics and science, such as linear algebra, vector spaces, and linear transformations.

Q: How do I find the coordinate vector of a vector with respect to a basis?

A: To find the coordinate vector of a vector with respect to a basis, you need to express the vector as a linear combination of the basis vectors. This involves solving a system of equations to find the coefficients of the linear combination.

Q: What is the difference between a coordinate vector and a vector?

A: A coordinate vector is a vector that represents the coefficients of the linear combination of the basis vectors that express a given vector. A vector is a mathematical object that has both magnitude and direction.

Q: Can I use any set of vectors as a basis?

A: No, not any set of vectors can be used as a basis. A basis must be a set of linearly independent vectors that span the vector space.

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

A: A set of vectors is linearly independent if none of the vectors can be expressed as a linear combination of the other vectors.

Q: What is the relationship between the coordinate vector and the basis vectors?

A: The coordinate vector is a linear combination of the basis vectors. The coefficients of the linear combination are the coordinates of the vector with respect to the basis.

Q: Can I change the basis of a vector space?

A: Yes, you can change the basis of a vector space. This is known as a change of basis transformation.

Q: How do I perform a change of basis transformation?

A: To perform a change of basis transformation, you need to find the matrix that represents the change of basis. This involves finding the coordinates of the basis vectors with respect to the new basis.

Q: What is the significance of the coordinate vector in linear algebra?

A: The coordinate vector is a fundamental concept in linear algebra. It provides a way to represent the coefficients of the linear combination of the basis vectors that express a given vector. This is useful in many areas of mathematics and science, such as linear transformations, matrix representations, and vector spaces.

Q: Can I use the coordinate vector to solve systems of linear equations?

A: Yes, you can use the coordinate vector to solve systems of linear equations. The coordinate vector provides a way to represent the coefficients of the linear combination of the basis vectors that express a given vector. This can be used to solve systems of linear equations.

Q: What are some common applications of coordinate vectors?

A: Coordinate vectors have many applications in mathematics and science, such as:

• Linear transformations
• Matrix representations
• Change of basis transformations
• Vector spaces
• Systems of linear equations

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

A: To choose a basis for a vector space, you need to select a set of linearly independent vectors that span the vector space. This can be done using various methods, such as the Gram-Schmidt process or the QR decomposition.

Q: What is the relationship between the coordinate vector and the vector space?

A: The coordinate vector is a vector that represents the coefficients of the linear combination of the basis vectors that express a given vector. The vector space is the set of all possible vectors that can be expressed as a linear combination of the basis vectors.

Q: Can I use the coordinate vector to find the dimension of a vector space?

A: Yes, you can use the coordinate vector to find the dimension of a vector space. The dimension of a vector space is the number of basis vectors required to span the vector space. The coordinate vector provides a way to represent the coefficients of the linear combination of the basis vectors that express a given vector. This can be used to find the dimension of the vector space.