https://file.aiquickdraw.com/mathbot/file/1726777153326uz98coiz.jpg

To solve this problem, we need to determine the change of basis matrix \( P \) from the standard basis \( \{ \mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3 \} \) to the given basis \( B = \left\{ \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 3 \end{pmatrix} \right\} \) in \( \mathbb{R}^3 \).

The matrix \( P \) is formed by expressing the basis vectors of \( B \) as columns in the matrix. So, \( P \) will be:

\[
P = \begin{pmatrix} 1 & 1 & 0 \\ 2 & 3 & 1 \\ 0 & 2 & 3 \end{pmatrix}
\]

This is the change of basis matrix from the standard basis \( \{ \mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3 \} \) to the basis \( B \).

Would you like further details on how to use this matrix to change coordinates between bases, or any specific questions on related topics?

Here are five questions to consider:
1. How do we use the matrix \( P \) to convert coordinates of a vector from one basis to another?
2. What would be the inverse of this change of basis matrix \( P \)?
3. How is the determinant of \( P \) related to the geometry of the transformation between bases?
4. How can we determine if a basis change preserves vector lengths and angles (orthogonality)?
5. What happens to the linear independence of vectors when performing a change of basis?

**Tip**: The inverse of the change of basis matrix \( P \) allows you to convert from the new basis back to the standard basis.

Consider the basis B = {(1,2,0), (1,3,2), (0,1,3)} of R^3. Find the change of basis matrix P from the standard basis {e1, e2, e3} to the basis B.

Learn how to find the change of basis matrix P from the standard basis {e1, e2, e3} to a new basis B = {(1,2,0), (1,3,2), (0,1,3)} in R^3. This problem involves key concepts in linear algebra, including matrix transformations and change of basis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation