Let
A=[4,-5,-3 5,-2,6.  -4, 5, 4.   8, -27, -47] and b=[-2 -47 -3 184]
Define the transformation T: R^3 -> R^4
 by T(x) = Ax
. Find a vector x
 whose image under T
 is b
.

 
 
.

x=[?,?,?]

To find a vector \( x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \) such that \( T(x) = Ax = b \), we need to solve the system of equations represented by \( Ax = b \).

Given:

\[
A = \begin{bmatrix} 
4 & -5 & -3 \\
5 & -2 & 6 \\
-4 & 5 & 4 \\
8 & -27 & -47 
\end{bmatrix}, \quad
b = \begin{bmatrix} 
-2 \\ -47 \\ -3 \\ 184 
\end{bmatrix}
\]

We need to solve the matrix equation:

\[
\begin{bmatrix} 
4 & -5 & -3 \\
5 & -2 & 6 \\
-4 & 5 & 4 \\
8 & -27 & -47 
\end{bmatrix} 
\begin{bmatrix} 
x_1 \\ x_2 \\ x_3 
\end{bmatrix}
=
\begin{bmatrix} 
-2 \\ -47 \\ -3 \\ 184 
\end{bmatrix}
\]

This system of linear equations can be solved using Gaussian elimination, or alternatively through matrix methods such as LU decomposition or using a computer algorithm. Let's proceed with solving it.The solution to the system \( Ax = b \) is:

\[
x = \begin{bmatrix} -3 \\ 1 \\ -5 \end{bmatrix}
\]

Thus, the vector \( x = [-3, 1, -5] \) is the one whose image under the transformation \( T(x) = Ax \) is \( b \).

Do you want more details on the solution process or have any questions?

Here are 5 related questions for further exploration:
1. How do we know that the solution for \( x \) is unique?
2. What is the geometric interpretation of the transformation \( T \)?
3. How can we verify the result by back-substitution?
4. What other methods can be used to solve this system?
5. How does the rank of matrix \( A \) affect the solution?

**Tip:** When solving a system of linear equations, using matrix decomposition methods (like LU or QR) can sometimes be more efficient than direct Gaussian elimination, especially for large matrices.

Let A = [[4,-5,-3], [5,-2,6], [-4,5,4], [8,-27,-47]] and b = [-2, -47, -3, 184]. Define the transformation T: R^3 -> R^4 by T(x) = Ax. Find a vector x whose image under T is b.

Explore how to solve the matrix equation T(x) = Ax where A is a 4x3 matrix and b is a 4-dimensional vector. Learn how Gaussian elimination or LU decomposition methods can be used to find the vector x, and understand the geometric interpretation of the transformation. Suitable for college-level students studying linear algebra.

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