To find the transition matrix from the basis $B$ to the basis $B'$, we need to express each vector in $B'$ as a linear combination of the vectors in $B$.

Given:

• Basis $B = \{(1, 0), (0, 1)\}$ (which is the standard basis).
• Basis $B' = \{(2, 8), (1, 5)\}$.

The transition matrix $P$ from $B$ to $B'$ can be constructed by writing each vector in $B'$ in terms of the basis vectors in $B$.

Step 1: Express each vector in $B'$ in terms of $B$

Each vector in $B'$ can be written as a linear combination of the vectors in $B$:

$(2, 8) = a(1, 0) + b(0, 1) = (a, b)$ Comparing components, we have: $a = 2, \quad b = 8$ So, $(2, 8) = 2(1, 0) + 8(0, 1)$.

$(1, 5) = c(1, 0) + d(0, 1) = (c, d)$ Comparing components, we have: $c = 1, \quad d = 5$ So, $(1, 5) = 1(1, 0) + 5(0, 1)$.

Step 2: Construct the transition matrix $P$

The coefficients we found form the columns of the transition matrix $P$ from $B$ to $B'$:

2 & 1 \\ 8 & 5 \end{bmatrix}$$ ### Conclusion The transition matrix from $$B$$ to $$B'$$ is: $$P = \begin{bmatrix} 2 & 1 \\ 8 & 5 \end{bmatrix}$$ Would you like a detailed explanation of how transition matrices work or have any questions? Here are some related questions for further exploration: 1. How do you find the inverse of a transition matrix? 2. What is the geometric interpretation of the transition matrix in a 2D plane? 3. How do you use a transition matrix to change coordinates from one basis to another? 4. What is the difference between the transition matrix from $$B$$ to $$B'$$ and from $$B'$$ to $$B$$? 5. How do you determine if a transition matrix is invertible? **Tip:** To convert coordinates of a vector in basis $$B'$$ back to basis $$B$$, use the inverse of the transition matrix $$P$$.