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The image shows a linear algebra problem. You are given the matrix \( A \) and the vector \( b \) as:

\[
A = \begin{pmatrix} -4 & 2 & -5 \\ 3 & -4 & 0 \\ -5 & 3 & -4 \end{pmatrix}, \quad b = \begin{pmatrix} -13 \\ -21 \\ -126 \end{pmatrix}
\]

The task is to find a vector \( x \) such that:

\[
T(x) = A x = b
\]

This involves solving the system of linear equations \( A x = b \). Let me solve this system and find the vector \( x \).The solution to the system \( A x = b \) is the vector:

\[
x = \begin{pmatrix} 163.93 \\ 128.20 \\ -77.27 \end{pmatrix}
\]

This is the vector \( x \) whose image under the transformation \( T(x) = A x \) is the given vector \( b \).

Would you like more details or have any other questions?

Here are five related questions you might find interesting:
1. How can we verify that \( A x = b \) holds true?
2. What is the determinant of matrix \( A \)?
3. What is the rank of the matrix \( A \)?
4. Can this system of equations be solved using Gaussian elimination?
5. How would you interpret the linear transformation geometrically?

**Tip**: Always check if a matrix is invertible before solving a system of equations, as it guarantees a unique solution.

Let A = [[-4, 2, -5], [3, -4, 0], [-5, 3, -4]] and b = [[-13], [-21], [-126]]. Define the transformation T: R^3 → R^3 by T(x) = A * x = b. Find a vector x whose image under T is b.

Learn how to solve the system of linear equations A * x = b, where A is a 3x3 matrix and b is a vector. This problem involves key linear algebra concepts such as matrix transformation and solving linear systems. The solution to A * x = b is provided step-by-step.

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