Questions: Consider the following two ordered bases of (R^3) : - (langle-1,1,-1rangle,langle-1,2,-1rangle,langle-2,2,-1rangle) - (langle-1,-1,1rangle,langle 1,2,-1rangle,langle 1,1,0rangle) a. Find the change of basis matrix from the basis to the basis. ([id]square^square=) [ beginarrayccc square square square square square square square square endarray ] b. Find the change of basis matrix from the basis to the basis. ([id]口^circ=) [ beginarrayccc square square square square square square square square endarray ]

Solution Steps

To find the change of basis matrix from one basis to another, we need to express each vector of the new basis in terms of the old basis. This involves setting up a system of linear equations and solving for the coefficients that express each new basis vector as a linear combination of the old basis vectors. The matrix formed by these coefficients is the change of basis matrix.

We have two ordered bases of \( \mathbb{R}^3 \): \[ \text{Basis 1} = \left\{ \langle -1, 1, -1 \rangle, \langle -1, 2, -1 \rangle, \langle -2, 2, -1 \rangle \right\} \] \[ \text{Basis 2} = \left\{ \langle -1, -1, 1 \rangle, \langle 1, 2, -1 \rangle, \langle 1, 1, 0 \rangle \right\} \]

To find the change of basis matrix \( [\mathrm{id}]_{\text{Basis 1}}^{\text{Basis 2}} \), we express the vectors of Basis 2 in terms of Basis 1. The resulting matrix is: \[ [\mathrm{id}]_{\text{Basis 1}}^{\text{Basis 2}} = \begin{bmatrix} -0 & 1 & -1 \\ 2 & 3 & -2 \\ 3 & 3 & -2 \end{bmatrix} \]

To find the change of basis matrix \( [\mathrm{id}]_{\text{Basis 2}}^{\text{Basis 1}} \), we express the vectors of Basis 1 in terms of Basis 2. The resulting matrix is: \[ [\mathrm{id}]_{\text{Basis 2}}^{\text{Basis 1}} = \begin{bmatrix} -0 & -1 & 1 \\ -2 & 3 & -2 \\ -3 & 3 & -2 \end{bmatrix} \]

Final Answer