Questions: Find the transition matrix from (B) to (B'). [ B=(4,-1),(3,2), B'=(1,0),(0,1) ] STEP 1: Begin by forming the following matrix. STEP 2: Determine the transition matrix.

Solution Steps

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

Given: \[ B = \{(4, -1), (3, 2)\} \] \[ B' = \{(1, 0), (0, 1)\} \]

Express each vector in \( B \) in terms of \( B' \):

• For \( (4, -1) \): \[ (4, -1) = 4(1, 0) + (-1)(0, 1) \] So, the coordinates are \( (4, -1) \) in \( B' \).

• For \( (3, 2) \): \[ (3, 2) = 3(1, 0) + 2(0, 1) \] So, the coordinates are \( (3, 2) \) in \( B' \).

Thus, the matrix [B':B] is: \[ [B':B] = \begin{pmatrix} 4 & 3 \\ -1 & 2 \end{pmatrix} \]

The transition matrix \( P \) from \( B \) to \( B' \) is the inverse of the matrix [B':B].

\[ P = ([B':B])^{-1} \]

To find the inverse of the matrix: \[ [B':B] = \begin{pmatrix} 4 & 3 \\ -1 & 2 \end{pmatrix} \]

The inverse of a 2x2 matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is: \[ \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]

For our matrix: \[ a = 4, b = 3, c = -1, d = 2 \] \[ ad - bc = (4)(2) - (3)(-1) = 8 + 3 = 11 \]

So, the inverse is: \[ \frac{1}{11} \begin{pmatrix} 2 & -3 \\ 1 & 4 \end{pmatrix} = \begin{pmatrix} \frac{2}{11} & \frac{-3}{11} \\ \frac{1}{11} & \frac{4}{11} \end{pmatrix} \]

Final Answer