Let T: R2 --> R2 be the linear transformation given by T(x,y)=(2x+y, x+y). Find T -1(x,y).

1. Break Down the Problem

We need to find the inverse transformation $T^{-1}(x, y)$ for the given linear transformation $T(x, y) = (2x + y, x + y)$.

2. Relevant Concepts

The function $T$ maps $\mathbb{R}^2$ to $\mathbb{R}^2$ and is defined as: $T(x,y) = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$ To find the inverse $T^{-1}$, we first need to find the inverse of the transformation matrix.

3. Analysis and Detail

1. Find the transformation matrix $A$: $A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$

2. Calculate the determinant of $A$: $\text{det}(A) = (2)(1) - (1)(1) = 2 - 1 = 1$

3. Find the inverse of matrix $A$ using the formula for the inverse of a $2 \times 2$ matrix: $A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} = \begin{pmatrix} 1 & -1 \\ -1 & 2 \end{pmatrix}$

4. Use the inverse matrix to find $T^{-1}(x, y)$: We have: $T^{-1}(x, y) = A^{-1} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 1 & -1 \\ -1 & 2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$ This results in: $T^{-1}(x, y) = \begin{pmatrix} x - y \\ -x + 2y \end{pmatrix}$

4. Verify and Summarize

Verifying, if we apply $T$ on $T^{-1}(x, y)$: $T(T^{-1}(x, y)) = T(x - y, -x + 2y) = \begin{pmatrix} 2(x - y) + (-x + 2y) \\ (x - y) + (-x + 2y) \end{pmatrix} = \begin{pmatrix} x + y \\ y \end{pmatrix}$ This confirms that we have derived the correct inverse transformation.

Final Answer

$T^{-1}(x, y) = (x - y, -x + 2y)$