Find the matrix of the linear transformation
F:R2→R3,[x,y]↦[2y−x,2x,5x−2y]

with respect to bases B={e⃗ 1,e⃗ 1+e⃗ 2}
 and C={e⃗ 1,e⃗ 2,e⃗ 3}
.

(Here {e⃗ 1,e⃗ 2}
 is the standard basis of R2
 and {e⃗ 1,e⃗ 2,e⃗ 3}
 is the standard basis of R3
.)

To find the matrix of the linear transformation \( F: \mathbb{R}^2 \to \mathbb{R}^3 \), given by:

\[
F([x, y]) = [2y - x, 2x, 5x - 2y]
\]

with respect to the basis \( B = \{ \mathbf{e}_1, \mathbf{e}_1 + \mathbf{e}_2 \} \) of \( \mathbb{R}^2 \) and the basis \( C = \{ \mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3 \} \) of \( \mathbb{R}^3 \), we proceed as follows:

### Step 1: Express the basis vectors of \( B \) in terms of the standard basis of \( \mathbb{R}^2 \).

The vectors in \( B \) are:
\[
\mathbf{v}_1 = \mathbf{e}_1 = [1, 0]
\]
\[
\mathbf{v}_2 = \mathbf{e}_1 + \mathbf{e}_2 = [1, 1]
\]

### Step 2: Apply the linear transformation \( F \) to each vector in \( B \).

1. Apply \( F \) to \( \mathbf{v}_1 = [1, 0] \):
   \[
   F([1, 0]) = [2(0) - 1, 2(1), 5(1) - 2(0)] = [-1, 2, 5]
   \]

2. Apply \( F \) to \( \mathbf{v}_2 = [1, 1] \):
   \[
   F([1, 1]) = [2(1) - 1, 2(1), 5(1) - 2(1)] = [1, 2, 3]
   \]

Thus, we have:
\[
F(\mathbf{v}_1) = [-1, 2, 5]
\]
\[
F(\mathbf{v}_2) = [1, 2, 3]
\]

### Step 3: Write these images as columns in the matrix with respect to the bases \( B \) and \( C \).

Since \( B \) has two vectors and \( C \) has three, the transformation matrix will be a \( 3 \times 2 \) matrix, with each column corresponding to the image of a basis vector from \( B \).

The matrix is:
\[
[F]_{B \to C} = \begin{bmatrix}
-1 & 1 \\
 2 & 2 \\
 5 & 3
\end{bmatrix}
\]

Thus, the matrix of the linear transformation \( F \) with respect to the bases \( B = \{\mathbf{e}_1, \mathbf{e}_1 + \mathbf{e}_2\} \) and \( C = \{\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3\} \) is:
\[
\begin{bmatrix}
-1 & 1 \\
 2 & 2 \\
 5 & 3
\end{bmatrix}
\]

Would you like further details or have any questions?

### Related Questions:
1. How do we generalize this method for arbitrary dimensions of linear transformations?
2. How do changes of basis affect the matrix representation of a linear transformation?
3. How can we verify if a matrix represents a linear transformation between different vector spaces?
4. What are the properties of a transformation matrix when moving between different bases?
5. How does the choice of basis affect the geometric interpretation of a transformation?

### Tip:
When working with different bases, always express the new basis vectors in terms of the standard basis first. This simplifies the process of finding transformation matrices.

Learn how to find the matrix of the linear transformation F: R^2 → R^3 given by F([x, y]) = [2y - x, 2x, 5x - 2y] with respect to specified bases B = {e1, e1 + e2} and C = {e1, e2, e3}. This problem involves concepts of linear algebra, matrix representation of transformations, and basis transformations.

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