The linear transformation $T: \mathbb{R}^2 \to \mathbb{R}^2$ is given as follows:

$T\left(\begin{bmatrix} v_1 \\ v_2 \end{bmatrix}\right) = \begin{bmatrix} 8v_1 + 2v_2 \\ 4v_1 + 6v_2 \end{bmatrix}$

We can express this transformation in matrix form. The matrix $A$ that represents the transformation $T$ is:

$A = \begin{bmatrix} 8 & 2 \\ 4 & 6 \end{bmatrix}$

Thus, the linear transformation can be written as:

$T(v) = A \cdot \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = \begin{bmatrix} 8 & 2 \\ 4 & 6 \end{bmatrix} \cdot \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}$

This gives the same result:

$T\left(\begin{bmatrix} v_1 \\ v_2 \end{bmatrix}\right) = \begin{bmatrix} 8v_1 + 2v_2 \\ 4v_1 + 6v_2 \end{bmatrix}$

Would you like more details on the properties of this transformation or its application?

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Related questions:

1. What is the determinant of matrix $A$, and how does it relate to the transformation?
2. How do you find the eigenvalues and eigenvectors of this matrix $A$?
3. What does the transformation do geometrically (e.g., scaling, rotation, etc.)?
4. How can we invert the transformation, and under what conditions is it invertible?
5. Can this transformation be composed with another to create a new transformation?

Tip:

Linear transformations can be visualized as matrix operations that alter the grid of vectors in $\mathbb{R}^2$, like stretching, compressing, or rotating.