Here
T : ℝ2 → ℝ2.
Fill in the blanks.
T(v)
=
T

v1

v2

=

8v1 + 2v2

4v1 + 6v2
=

v

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.

T : ℝ² → ℝ². Fill in the blanks. T(v) = T [v1, v2] = [8v1 + 2v2, 4v1 + 6v2] = v

Explore the linear transformation T(v) = [8v1 + 2v2, 4v1 + 6v2] in ℝ². This problem covers key concepts in linear algebra, including matrix multiplication and linear transformations. Learn how to represent transformations with matrices. Suitable for students in Grades 11-12 and college-level courses.

Math Problem Statement

Here T : ℝ2 → ℝ2. Fill in the blanks. T(v)

4v1 + 6v2

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