Question: Let $e_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $e_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$, $y_1 = \begin{bmatrix} 2 \\ 7 \end{bmatrix}$, and $y_2 = \begin{bmatrix} -2 \\ 8 \end{bmatrix}$, and let $T: \mathbb{R}^2 \to \mathbb{R}^2$ be a linear transformation that maps $e_1$ into $y_1$ and maps $e_2$ into $y_2$. Find the images of $\begin{bmatrix} 5 \\ -3 \end{bmatrix}$ and $\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$. The image of $\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ is
1. **State the problem:** We have a linear transformation $T: \mathbb{R}^2 \to \mathbb{R}^2$ defined by its action on the standard basis vectors:
$$T(e_1) = y_1 = \begin{bmatrix} 2 \\ 7 \end{bmatrix}, \quad T(e_2) = y_2 = \begin{bmatrix} -2 \\ 8 \end{bmatrix}$$
We want to find the images under $T$ of the vectors
$$v_1 = \begin{bmatrix} 5 \\ -3 \end{bmatrix} \quad \text{and} \quad v = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}.$$
2. **Recall the formula for linear transformations:** For any vector $v = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ in $\mathbb{R}^2$, since $T$ is linear and $e_1, e_2$ form a basis, we have
$$T(v) = T(x_1 e_1 + x_2 e_2) = x_1 T(e_1) + x_2 T(e_2).$$
3. **Apply the formula to $v_1$:**
$$T\left(\begin{bmatrix} 5 \\ -3 \end{bmatrix}\right) = 5 T(e_1) + (-3) T(e_2) = 5 \begin{bmatrix} 2 \\ 7 \end{bmatrix} - 3 \begin{bmatrix} -2 \\ 8 \end{bmatrix}.$$
4. **Calculate each term:**
$$5 \begin{bmatrix} 2 \\ 7 \end{bmatrix} = \begin{bmatrix} 10 \\ 35 \end{bmatrix}, \quad -3 \begin{bmatrix} -2 \\ 8 \end{bmatrix} = \begin{bmatrix} 6 \\ -24 \end{bmatrix}.$$
5. **Add the vectors:**
$$\begin{bmatrix} 10 \\ 35 \end{bmatrix} + \begin{bmatrix} 6 \\ -24 \end{bmatrix} = \begin{bmatrix} 16 \\ 11 \end{bmatrix}.$$
So,
$$T\left(\begin{bmatrix} 5 \\ -3 \end{bmatrix}\right) = \begin{bmatrix} 16 \\ 11 \end{bmatrix}.$$
6. **Find the image of a general vector $v = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$:**
$$T\left(\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\right) = x_1 T(e_1) + x_2 T(e_2) = x_1 \begin{bmatrix} 2 \\ 7 \end{bmatrix} + x_2 \begin{bmatrix} -2 \\ 8 \end{bmatrix}.$$
7. **Write the result explicitly:**
$$= \begin{bmatrix} 2 x_1 \\ 7 x_1 \end{bmatrix} + \begin{bmatrix} -2 x_2 \\ 8 x_2 \end{bmatrix} = \begin{bmatrix} 2 x_1 - 2 x_2 \\ 7 x_1 + 8 x_2 \end{bmatrix}.$$
**Final answers:**
- $T\left(\begin{bmatrix} 5 \\ -3 \end{bmatrix}\right) = \begin{bmatrix} 16 \\ 11 \end{bmatrix}$
- $T\left(\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\right) = \begin{bmatrix} 2 x_1 - 2 x_2 \\ 7 x_1 + 8 x_2 \end{bmatrix}$