https://file.aiquickdraw.com/mathbot/file/172477044731200ig1pwk.jpg

To find the range (or column space) of the matrix using its echelon form, we'll follow these steps:

1. **Write down the matrix**: The matrix from the image is:
   \[
   A = \begin{pmatrix}
   1 & 2 & 3 & 4 \\
   2 & 1 & 4 & 3 \\
   3 & 0 & 5 & -10
   \end{pmatrix}
   \]

2. **Convert the matrix to Row Echelon Form (REF)** using Gaussian elimination.

Let's perform the row operations:

\[
\text{Matrix } A = \begin{pmatrix}
1 & 2 & 3 & 4 \\
2 & 1 & 4 & 3 \\
3 & 0 & 5 & -10
\end{pmatrix}
\]

- Subtract 2 times Row 1 from Row 2:
  \[
  R_2 = R_2 - 2R_1 \quad \Rightarrow \quad \begin{pmatrix}
  1 & 2 & 3 & 4 \\
  0 & -3 & -2 & -5 \\
  3 & 0 & 5 & -10
  \end{pmatrix}
  \]

- Subtract 3 times Row 1 from Row 3:
  \[
  R_3 = R_3 - 3R_1 \quad \Rightarrow \quad \begin{pmatrix}
  1 & 2 & 3 & 4 \\
  0 & -3 & -2 & -5 \\
  0 & -6 & -4 & -22
  \end{pmatrix}
  \]

- To simplify Row 3, add 2 times Row 2 to Row 3:
  \[
  R_3 = R_3 + 2R_2 \quad \Rightarrow \quad \begin{pmatrix}
  1 & 2 & 3 & 4 \\
  0 & -3 & -2 & -5 \\
  0 & 0 & 0 & -32
  \end{pmatrix}
  \]

Now, the matrix is in row echelon form. 

3. **Identify the pivot columns**: The pivot positions are in the first, second, and fourth columns. The corresponding pivot columns in the original matrix are:

   \[
   \text{Columns:} \quad \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \quad \begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix}, \quad \begin{pmatrix} 4 \\ 3 \\ -10 \end{pmatrix}
   \]

4. **Determine the range**: The range (column space) of the matrix is spanned by the pivot columns of the original matrix. So, the basis for the range is:
   \[
   \text{Basis:} \quad \left\{ \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 4 \\ 3 \\ -10 \end{pmatrix} \right\}
   \]

5. **Conclusion**: The range of the matrix is the span of the vectors \(\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}\), \(\begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix}\), and \(\begin{pmatrix} 4 \\ 3 \\ -10 \end{pmatrix}\).

Would you like more details on the steps or have any further questions?

Here are some related questions to consider:
1. How do you determine the rank of a matrix?
2. What is the significance of the row space of a matrix?
3. How does the dimension of the range relate to the rank of the matrix?
4. How can you find the null space of a matrix?
5. What is the significance of a matrix's determinant in relation to its rank?

**Tip**: When performing Gaussian elimination, always keep track of the operations to easily backtrack if a mistake is made.

Learn how to find the range of a matrix using its echelon form with a detailed step-by-step guide. This problem involves matrix operations and linear algebra concepts, focusing on identifying the column space from the row echelon form. Suitable for advanced college-level students and beyond.

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