Find the coordinate vector [Bold x ]Subscript Upper B of x relative to the given basis B
The Correct Answer and Explanation is:
To find the coordinate vector ([ \mathbf{x} ]_B) of a vector (\mathbf{x}) relative to a given basis (B), we need to express (\mathbf{x}) as a linear combination of the basis vectors in (B). This process involves several steps that are crucial for understanding how to relate any vector to a specified basis.
Steps to Find ([ \mathbf{x} ]_B):
- Identify the Basis (B): Let’s assume the basis (B) consists of vectors (\mathbf{b}_1, \mathbf{b}_2, \ldots, \mathbf{b}_n). Each of these vectors should be linearly independent.
- Express (\mathbf{x}) as a Linear Combination: We want to express the vector (\mathbf{x}) in the form:
[
\mathbf{x} = c_1 \mathbf{b}_1 + c_2 \mathbf{b}_2 + \ldots + c_n \mathbf{b}_n
]
where (c_1, c_2, \ldots, c_n) are the coefficients we need to find. These coefficients represent the components of the coordinate vector ([ \mathbf{x} ]_B). - Set Up the Equation: Rearranging the equation gives us:
[
\mathbf{x} – c_1 \mathbf{b}_1 – c_2 \mathbf{b}_2 – \ldots – c_n \mathbf{b}_n = \mathbf{0}
]
This represents a system of linear equations based on the components of the vectors involved. - Formulate the Matrix Equation: This system can be written in matrix form as (A \mathbf{c} = \mathbf{x}), where (A) is the matrix whose columns are the basis vectors (B) and (\mathbf{c}) is the vector of coefficients ([c_1, c_2, \ldots, c_n]^T).
- Solve for the Coefficients: Use techniques such as Gaussian elimination, matrix inversion, or any suitable method to solve for (\mathbf{c}).
- Construct the Coordinate Vector: Once you have the coefficients (c_1, c_2, \ldots, c_n), the coordinate vector is given by:
[
[ \mathbf{x} ]_B = \begin{bmatrix} c_1 \ c_2 \ \vdots \ c_n \end{bmatrix}
]
Example:
Suppose (\mathbf{x} = \begin{bmatrix} 3 \ 4 \end{bmatrix}) and the basis (B = { \begin{bmatrix} 1 \ 0 \end{bmatrix}, \begin{bmatrix} 0 \ 1 \end{bmatrix} }).
The equations would be:
[
\begin{bmatrix} 3 \ 4 \end{bmatrix} = c_1 \begin{bmatrix} 1 \ 0 \end{bmatrix} + c_2 \begin{bmatrix} 0 \ 1 \end{bmatrix}
]
From here, we can see that (c_1 = 3) and (c_2 = 4), leading to:
[
[ \mathbf{x} ]_B = \begin{bmatrix} 3 \ 4 \end{bmatrix}
]
Conclusion
The coordinate vector ([ \mathbf{x} ]_B) gives us the representation of the vector (\mathbf{x}) in the context of the basis (B), which is essential in various applications such as computer graphics, physics, and advanced mathematics, as it allows us to manipulate and understand vectors within different reference frames.