College Algebra

Linear Algebra

To find the matrix representation \([T]_{B_2}^{B_1}\) of the linear transformation \( T: \mathbb{R}^3 \to \mathbb{R}^2 \) relative to the bases \( B_1 \) and \( B_2 \), we follow these steps: ### Step 1: Apply \( T \) to Each Basis Vector of \( B_1 \) The basis \( B_1 \) for \( \mathbb{R}^3 \) is given by \( B_1 = \{(1,1,0), (0,1,1), (0,0,1)\} \). We need to compute \( T \) on each of these vectors. 1. **For \( (1,1,0) \):** \[ T(1,1,0) = [4 \cdot 1 + 1 \cdot 1 - 0 \cdot 0, 1 \cdot 1 - 1 \cdot 1 + 0 \cdot 0] = [5, 0] \] 2. **For \( (0,1,1) \):** \[ T(0,1,1) = [4 \cdot 0 + 1 \cdot 1 - 1 \cdot 1, 0 \cdot 1 - 1 \cdot 1 + 1 \cdot 1] = [0, 0] \] 3. **For \( (0,0,1) \):** \[ T(0,0,1) = [4 \cdot 0 + 0 \cdot 0 - 1 \cdot 1, 0 \cdot 0 - 0 \cdot 1 + 1 \cdot 1] = [-1, 1] \] ### Step 2: Express Each Result as a Linear Combination of \( B_2 \) The basis \( B_2 \) for \( \mathbb{R}^2 \) is given by \( B_2 = \{(1,4), (1,5)\} \). We need to express each vector obtained from Step 1 as a linear combination of these basis vectors. 1. **For \( [5, 0] \):** We solve the system: \[ a(1,4) + b(1,5) = (5,0) \] This gives the equations: \[ \begin{align*} a + b &= 5 \\ 4a + 5b &= 0 \end{align*} \] Solving this system, we find \( a = 5 \) and \( b = -4 \). 2. **For \( [0, 0] \):** \[ a(1,4) + b(1,5) = (0,0) \] This gives: \[ \begin{align*} a + b &= 0 \\ 4a + 5b &= 0 \end{align*} \] Solving this system, we find \( a = 0 \) and \( b = 0 \). 3. **For \( [-1, 1] \):** \[ a(1,4) + b(1,5) = (-1,1) \] This gives: \[ \begin{align*} a + b &= -1 \\ 4a + 5b &= 1 \end{align*} \] Solving this system, we find \( a = -3 \) and \( b = 2 \). ### Step 3: Construct the Matrix \([T]_{B_2}^{B_1}\) The columns of \([T]_{B_2}^{B_1}\) are the coordinates of the transformed vectors (from Step 2) relative to \( B_2 \). Thus, the matrix representation is: \[ [T]_{B_2}^{B_1} = \begin{bmatrix} 5 & 0 & -3 \\ -4 & 0 & 2 \end{bmatrix} \] This matrix represents the linear transformation \( T \) relative to the bases \( B_1 \) and \( B_2 \).

Math

Let \( T: \mathbb{R}^3 \to \mathbb{R}^2 \) be the linear transformation defined by \[ T[x, y, z] = [4x + y - z, x - y + z]. \] Compute the representation matrix of \( T \) relative to the basis \[ B_1 = \{(1,1,0), (0,1,1), (0,0,1)\} \text{ of } \mathbb{R}^3 \text{ and } \] \[ B_2 = \{(1,4), (1,5)\} \text{ of } \mathbb{R}^2. \] I.e., find \([T]_{B_2}^{B_1}\).

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