Questions: Find the change of basis matrix from 𝓑₁ to 𝓑₂. 𝓑₁= [[7, 4, 2], [6, 0, -6], [-3, 3, 2]] 𝓑₂= [[7, -2, -1], [4, 3, 9], [-8, 1, 5]]

Find the change of basis matrix from 𝓑₁ to 𝓑₂. 𝓑₁= [[7, 4, 2], [6, 0, -6], [-3, 3, 2]] 𝓑₂= [[7, -2, -1], [4, 3, 9], [-8, 1, 5]]
Transcript text: Find the change of basis matrix from $\mathcal{B}_{1}$ to $\mathcal{B}_{2}$. \[ \begin{array}{l} \mathcal{B}_{1}=\left\{\left[\begin{array}{l} 7 \\ 4 \\ 2 \end{array}\right],\left[\begin{array}{r} 6 \\ 0 \\ -6 \end{array}\right],\left[\begin{array}{r} -3 \\ 3 \\ 2 \end{array}\right]\right\} \\ \mathcal{B}_{2}=\left\{\left[\begin{array}{r} 7 \\ -2 \\ -1 \end{array}\right],\left[\begin{array}{l} 4 \\ 3 \\ 9 \end{array}\right],\left[\begin{array}{r} -8 \\ 1 \\ 5 \end{array}\right]\right\} \end{array} \]
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Solution

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Solution Steps

To find the change of basis matrix from \(\mathcal{B}_{1}\) to \(\mathcal{B}_{2}\), we need to express each vector in \(\mathcal{B}_{1}\) as a linear combination of the vectors in \(\mathcal{B}_{2}\). This involves solving a system of linear equations for each vector in \(\mathcal{B}_{1}\) to find its coordinates in terms of \(\mathcal{B}_{2}\). The resulting coordinates form the columns of the change of basis matrix.

Step 1: Define the Basis Vectors

We are given two sets of basis vectors: \[ \mathcal{B}_{1} = \left\{ \begin{bmatrix} 7 \\ 4 \\ 2 \end{bmatrix}, \begin{bmatrix} 6 \\ 0 \\ -6 \end{bmatrix}, \begin{bmatrix} -3 \\ 3 \\ 2 \end{bmatrix} \right\} \] \[ \mathcal{B}_{2} = \left\{ \begin{bmatrix} 7 \\ -2 \\ -1 \end{bmatrix}, \begin{bmatrix} 4 \\ 3 \\ 9 \end{bmatrix}, \begin{bmatrix} -8 \\ 1 \\ 5 \end{bmatrix} \right\} \]

Step 2: Form the Matrix Representations

The matrix representation of \(\mathcal{B}_{1}\) and \(\mathcal{B}_{2}\) are: \[ B_1 = \begin{bmatrix} 7 & 6 & -3 \\ 4 & 0 & 3 \\ 2 & -6 & 2 \end{bmatrix} \] \[ B_2 = \begin{bmatrix} 7 & 4 & -8 \\ -2 & 3 & 1 \\ -1 & 9 & 5 \end{bmatrix} \]

Step 3: Compute the Inverse of \(B_2\)

To find the change of basis matrix from \(\mathcal{B}_{1}\) to \(\mathcal{B}_{2}\), we first compute the inverse of \(B_2\): \[ B_2^{-1} = \begin{bmatrix} 0.0303 & -0.4646 & 0.1414 \\ 0.0455 & 0.1364 & 0.0455 \\ -0.0758 & -0.3384 & 0.1465 \end{bmatrix} \]

Step 4: Calculate the Change of Basis Matrix

The change of basis matrix from \(\mathcal{B}_{1}\) to \(\mathcal{B}_{2}\) is given by: \[ P_{\mathcal{B}_{1} \to \mathcal{B}_{2}} = B_2^{-1} B_1 \] \[ P_{\mathcal{B}_{1} \to \mathcal{B}_{2}} = \begin{bmatrix} -1.364 & -0.6667 & -1.202 \\ 0.9545 & -6.939 \times 10^{-17} & 0.3636 \\ -1.591 & -1.333 & -0.4949 \end{bmatrix} \]

Final Answer

The change of basis matrix from \(\mathcal{B}_{1}\) to \(\mathcal{B}_{2}\) is: \[ \boxed{\begin{bmatrix} -1.364 & -0.6667 & -1.202 \\ 0.9545 & -6.939 \times 10^{-17} & 0.3636 \\ -1.591 & -1.333 & -0.4949 \end{bmatrix}} \]

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