How to Calculate a Change of Basis Matrix Step-by-Step

📚 Understanding Change of Basis Matrices

In linear algebra, a change of basis matrix allows us to express the coordinates of a vector in one basis in terms of another basis. This is fundamental when working with different coordinate systems or when simplifying linear transformations. Think of it as translating between different languages – the underlying object (the vector) remains the same, but the way we describe it changes.

📜 Historical Context

The concept of change of basis arose from the need to analyze linear transformations and vector spaces from different perspectives. Early mathematicians like Arthur Cayley and Hermann Grassmann laid the groundwork for linear algebra, which later formalized the change of basis concept. It's a powerful tool that simplifies many complex problems in physics, engineering, and computer science.

🔑 Key Principles

• 🔑Basis: A set of linearly independent vectors that span a vector space. Any vector in the space can be written as a linear combination of these basis vectors.
• ➕Linear Combination: Expressing a vector as the sum of scalar multiples of other vectors.
• 🔄Change of Basis: Finding the matrix that transforms the coordinates of a vector from one basis to another.
• 📏Coordinate Vector: A vector representing the coefficients used in a linear combination of basis vectors to express a given vector.

📝 Step-by-Step Calculation

Here's how to calculate a change of basis matrix:

1. ✔️Identify the two bases: Let $B = \{v_1, v_2, ..., v_n\}$ be the original basis and $B' = \{u_1, u_2, ..., u_n\}$ be the new basis.
2. ✍️Express the new basis vectors in terms of the original basis: For each $u_i$ in $B'$, find the coefficients $a_{1i}, a_{2i}, ..., a_{ni}$ such that $u_i = a_{1i}v_1 + a_{2i}v_2 + ... + a_{ni}v_n$.
3. 🧮Form the change of basis matrix: The change of basis matrix $P$ from $B'$ to $B$ has the coefficients $a_{ij}$ as its columns. That is, $$P = \begin{bmatrix} a_{11} & a_{12} & ... & a_{1n} \\ a_{21} & a_{22} & ... & a_{2n} \\ ... & ... & ... & ... \\ a_{n1} & a_{n2} & ... & a_{nn} \end{bmatrix}$$ This matrix transforms the coordinates of a vector in the $B'$ basis to its coordinates in the $B$ basis.

🧮 Example 1: $\mathbb{R}^2$

Let $B = \{\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}\}$ (the standard basis) and $B' = \{\begin{bmatrix} 1 \\ 1 \end{bmatrix}, \begin{bmatrix} -1 \\ 1 \end{bmatrix}\}$. Find the change of basis matrix from $B'$ to $B$.

Solution:

• ✔️Express $B'$ vectors in terms of $B$: $$\begin{bmatrix} 1 \\ 1 \end{bmatrix} = 1 \begin{bmatrix} 1 \\ 0 \end{bmatrix} + 1 \begin{bmatrix} 0 \\ 1 \end{bmatrix}$$ $$\begin{bmatrix} -1 \\ 1 \end{bmatrix} = -1 \begin{bmatrix} 1 \\ 0 \end{bmatrix} + 1 \begin{bmatrix} 0 \\ 1 \end{bmatrix}$$
• ✍️Form the change of basis matrix: $$P = \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix}$$

🧑‍🏫 Example 2: Another $\mathbb{R}^2$ Example

Let $B = \{\begin{bmatrix} 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \end{bmatrix}\}$ and $B' = \{\begin{bmatrix} 3 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \end{bmatrix}\}$. Find the change-of-basis matrix from $B'$ to $B$.

Solution:

• ✔️Express $B'$ vectors in terms of $B$: We need to find coefficients $a, b, c, d$ such that: $$\begin{bmatrix} 3 \\ 2 \end{bmatrix} = a \begin{bmatrix} 2 \\ 1 \end{bmatrix} + b \begin{bmatrix} 1 \\ 1 \end{bmatrix}$$ $$\begin{bmatrix} 1 \\ 0 \end{bmatrix} = c \begin{bmatrix} 2 \\ 1 \end{bmatrix} + d \begin{bmatrix} 1 \\ 1 \end{bmatrix}$$ Solving these systems of equations, we get: $a = 1, b = 1$ and $c = -1, d = 2$.
• ✍️Form the change of basis matrix: $$P = \begin{bmatrix} 1 & -1 \\ 1 & 2 \end{bmatrix}$$

💡 Tips and Tricks

• 🧮Matrix Inversion: To find the change of basis matrix from $B$ to $B'$, calculate the inverse of the change of basis matrix from $B'$ to $B$.
• ✔️Verify: Always verify your result by applying the change of basis matrix to a known coordinate vector and checking if the resulting vector is correct in the original basis.
• 🧭Orientation: Be mindful of the order of the bases. The change of basis matrix from $B$ to $B'$ is different from the change of basis matrix from $B'$ to $B$.

🌍 Real-world Applications

• 💻Computer Graphics: Used for transforming objects between different coordinate systems.
• ⚙️Robotics: Essential for calculating the position and orientation of robot arms.
• 🛰️Satellite Navigation: Used in GPS systems to transform coordinates between different reference frames.

📝 Practice Quiz

1. ❓ Let $B = \{\begin{bmatrix} 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}\}$ and $B' = \{\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \end{bmatrix}\}$. Find the change of basis matrix from $B'$ to $B$.
2. ❓ Let $B = \{\begin{bmatrix} 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \end{bmatrix}\}$ and $B' = \{\begin{bmatrix} 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \end{bmatrix}\}$. Find the change of basis matrix from $B'$ to $B$.
3. ❓ Let $B = \{\begin{bmatrix} 1 \\ 1 \end{bmatrix}, \begin{bmatrix} -1 \\ 1 \end{bmatrix}\}$ and $B' = \{\begin{bmatrix} 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \end{bmatrix}\}$. Find the change of basis matrix from $B'$ to $B$.
4. ❓ Let $B = \{\begin{bmatrix} 3 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \end{bmatrix}\}$ and $B' = \{\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}\}$. Find the change of basis matrix from $B'$ to $B$.
5. ❓ Let $B = \{\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}\}$ and $B' = \{\begin{bmatrix} 2 \\ 3 \end{bmatrix}, \begin{bmatrix} -1 \\ 1 \end{bmatrix}\}$. Find the change of basis matrix from $B'$ to $B$.
6. ❓ Let $B = \{\begin{bmatrix} 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \end{bmatrix}\}$ and $B' = \{\begin{bmatrix} 3 \\ 3 \end{bmatrix}, \begin{bmatrix} -1 \\ 1 \end{bmatrix}\}$. Find the change of basis matrix from $B'$ to $B$.
7. ❓ Let $B = \{\begin{bmatrix} 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}\}$ and $B' = \{\begin{bmatrix} 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \end{bmatrix}\}$. Find the change of basis matrix from $B'$ to $B$.

🎓 Conclusion

Calculating a change of basis matrix might seem complex at first, but with a clear understanding of the underlying principles and a step-by-step approach, it becomes manageable. This powerful tool is essential for simplifying problems in various fields and provides a deeper understanding of linear transformations. Keep practicing, and you'll master it in no time!