Questions: Find the B-matrix for the transformation x ↦ A x, where B=b1, b2. A=[ [ -6 1 ] [ 3 1 ] ], b1=[ [ -1 ] [ -1 ] ], b2=[ [ 1 ] [ 2 ] ] The B-matrix of the given transformation is .

Transcript text: Find the $B$-matrix for the transformation $\mathbf{x} \mapsto \mathrm{A} \mathbf{x}$, where $B=\left\{\mathbf{b}_{1}, \mathbf{b}_{2}\right\}$. \[ A=\left[\begin{array}{rr} -6 & 1 \\ 3 & 1 \end{array}\right], \quad \mathbf{b}_{1}=\left[\begin{array}{l} -1 \\ -1 \end{array}\right], \quad \mathbf{b}_{2}=\left[\begin{array}{l} 1 \\ 2 \end{array}\right] \] The $B$-matrix of the given transformation is $\square$ .

Solution

Solution Steps

Step 1: Define the Transformation Matrix and Basis Vectors

We are given the transformation matrix $ A = \begin{bmatrix} -6 & 1 \\ 3 & 1 \end{bmatrix} $ and the basis vectors $ \mathbf{b}_{1} = \begin{bmatrix} -1 \\ -1 \end{bmatrix} $ and $ \mathbf{b}_{2} = \begin{bmatrix} 1 \\ 2 \end{bmatrix} $.

Step 2: Construct the Basis Matrix

We construct the basis matrix $ B $ using the basis vectors: \[ B = \begin{bmatrix} -1 & 1 \\ -1 & 2 \end{bmatrix} \]

Step 3: Calculate the Inverse of the Basis Matrix

Next, we compute the inverse of the basis matrix $ B $: \[ B^{-1} = \begin{bmatrix} -2 & 1 \\ -1 & 1 \end{bmatrix} \]

Step 4: Compute the B-Matrix

Finally, we find the $ B $-matrix for the transformation by multiplying the inverse of the basis matrix $ B^{-1} $ with the transformation matrix $ A $: \[ B\text{-matrix} = B^{-1} A = \begin{bmatrix} -2 & 1 \\ -1 & 1 \end{bmatrix} \begin{bmatrix} -6 & 1 \\ 3 & 1 \end{bmatrix} = \begin{bmatrix} 15 & -1 \\ 9 & 0 \end{bmatrix} \]