### Understanding Linear Transformations: Finding the Standard Matrix**Problem Statement:**Assume that \( T \) is a linear transformation. Find the standard matrix of \( T \).\[ T: \mathbb{R}^2 \to \mathbb{R}^2 \] This transformation is a vertical shear transformation that maps \( \mathbf{e}_1 \) into \( \mathbf{e}_1 - 3\mathbf{e}_2 \), but leaves the vector \( \mathbf{e}_2 \) unchanged.**Instructions:**Type an integer or simplified fraction for each matrix element.---**Explanation:**To find the standard matrix \( A \) for this transformation, we consider how the basis vectors are transformed.1. **Effect on Basis Vector \( \mathbf{e}_1 \):** - Originally: \( \mathbf{e}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \) - Transformed: \( \mathbf{e}_1 - 3\mathbf{e}_2 = \begin{bmatrix} 1 \\ -3 \end{bmatrix} \)2. **Effect on Basis Vector \( \mathbf{e}_2 \):** - Remains unchanged: \( \mathbf{e}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \)Thus, the standard matrix \( A \) is formed by these transformed vectors:\[ A = \begin{bmatrix} 1 & 0 \\ -3 & 1 \end{bmatrix} \]**Conclusion:**The standard matrix \( A \) of the given linear transformation can be used to apply this transformation to any vector in \( \mathbb{R}^2 \).

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Assume that T is a linear transformation. Find the standard matrix of T. T: R² R² is a vertical shear transformation that maps e, into e, -3e2 but leaves the vector e2 unchanged. ... A = (Type an integer or simplified fraction for each matrix element.)

Assume that T is a linear transformation. Find the standard matrix of T. T: R² R² is a vertical shear transformation that maps e, into e, -3e2 but leaves the vector e2 unchanged. ... A = (Type an integer or simplified fraction for each matrix element.)

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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### Understanding Linear Transformations: Finding the Standard Matrix

**Problem Statement:**

Assume that \( T \) is a linear transformation. Find the standard matrix of \( T \).

\[ T: \mathbb{R}^2 \to \mathbb{R}^2 \]

This transformation is a vertical shear transformation that maps \( \mathbf{e}_1 \) into \( \mathbf{e}_1 - 3\mathbf{e}_2 \), but leaves the vector \( \mathbf{e}_2 \) unchanged.

**Instructions:**

Type an integer or simplified fraction for each matrix element.

---

**Explanation:**

To find the standard matrix \( A \) for this transformation, we consider how the basis vectors are transformed.

1. **Effect on Basis Vector \( \mathbf{e}_1 \):**
- Originally: \( \mathbf{e}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \)
- Transformed: \( \mathbf{e}_1 - 3\mathbf{e}_2 = \begin{bmatrix} 1 \\ -3 \end{bmatrix} \)

2. **Effect on Basis Vector \( \mathbf{e}_2 \):**
- Remains unchanged: \( \mathbf{e}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \)

Thus, the standard matrix \( A \) is formed by these transformed vectors:

\[ A = \begin{bmatrix} 1 & 0 \\ -3 & 1 \end{bmatrix} \]

**Conclusion:**

The standard matrix \( A \) of the given linear transformation can be used to apply this transformation to any vector in \( \mathbb{R}^2 \).

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