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I have displayed the image, and it appears to contain a mathematical question related to functional analysis or linear algebra.

Let me extract the text and provide you with a solution.The problem in the image seems to ask about a matrix \( A \) and finding the range of the transformation \( T \) defined by \( T(x) = Ax \). Let's summarize the key points:

"The transformation \( T \) is from \( \mathbb{R}^4 \) to \( \mathbb{R}^3 \) and the matrix \( A \) is given as \( A = \begin{pmatrix} 1 & -1 & 0 & 1 \\ 0 & 1 & -1 & 0 \\ 1 & -1 & 0 & 1 \end{pmatrix} \). You are tasked with finding the range of \( T(x) \)."

### Steps to Solve:
1. **Matrix Dimensions**: The matrix \( A \) is of size \( 3 \times 4 \), meaning it maps a 4-dimensional vector \( x \in \mathbb{R}^4 \) to a 3-dimensional vector \( T(x) \in \mathbb{R}^3 \).
   
2. **Range of Transformation**: To find the range, we need to find the column space of matrix \( A \). This is done by performing Gaussian elimination on \( A \) to find the independent columns (basis vectors).

I'll calculate the reduced row echelon form (RREF) of matrix \( A \) to determine the basis of the range.The reduced row echelon form (RREF) of the matrix \( A \) is:

\[
\begin{pmatrix}
1 & 0 & 1 \\
0 & 1 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}
\]

The pivot columns are 0 and 1, which means that the first and second columns of the original matrix \( A \) form a basis for the range of \( T \).

### Conclusion:
The range of the transformation \( T \) is spanned by the first and second columns of matrix \( A \). Thus, the range of \( T(x) \) is:

\[
\text{Range}(T) = \text{span}\left(\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix} -1 \\ 1 \\ -1 \end{pmatrix}\right)
\]

This is a subspace of \( \mathbb{R}^3 \) with dimension 2.

Would you like further details or have any questions? 

Here are five follow-up questions to consider:
1. What does it mean for a set of vectors to span a space?
2. How can the rank of a matrix help us understand the dimensions of its range?
3. What is the null space of this matrix, and how can it be found?
4. How do linear transformations relate to matrix multiplication?
5. What is the significance of the pivot columns in Gaussian elimination?

**Tip**: When finding the range of a transformation, always look for the pivot columns in the matrix after performing Gaussian elimination. These indicate the independent columns that span the range.

The transformation T(x) = Ax, where A is a 3x4 matrix, maps vectors from R^4 to R^3. Find the range of T(x) = A(x).

Learn how to find the range of the transformation T(x) = A(x), where A is a 3x4 matrix. This problem involves performing Gaussian elimination to identify the independent columns that span the range of the transformation. The solution highlights key concepts in linear algebra, including matrix transformations, column spaces, and the rank-nullity theorem. Suitable for undergraduate students.

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