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Let’s go through the problems one by one and solve them in detail.

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### **(1) Column space and nullspace of \( A \):**

We are given the matrix:
\[
A = \begin{bmatrix} 
2 & 4 & 6 & 4 \\ 
2 & 5 & 7 & 6 \\ 
2 & 3 & 5 & 2 
\end{bmatrix}.
\]

1. **Column Space**:  
   The column space of \( A \) is the span of its columns. To determine the column space, we reduce \( A \) to its column echelon form (or row echelon form) using Gaussian elimination to identify the pivot columns.

2. **Nullspace**:  
   The nullspace of \( A \) is the set of all solutions to the homogeneous equation \( A \mathbf{x} = \mathbf{0} \). To find it, we solve \( A \mathbf{x} = \mathbf{0} \) using row-reduction.

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**Finding the complete solution to \( A \mathbf{x} = \mathbf{b} \) with \( \mathbf{b} = \begin{bmatrix} 4 \\ 3 \\ 5 \end{bmatrix} \):**

We solve the system \( A \mathbf{x} = \mathbf{b} \) by reducing the augmented matrix \( [A \, | \, \mathbf{b}] \). This involves row reduction to find a particular solution \( \mathbf{x}_p \) and a general solution \( \mathbf{x} = \mathbf{x}_p + \mathbf{x}_n \), where \( \mathbf{x}_n \) is the nullspace solution.

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### **(2) Complete solution in the form \( \mathbf{x} = \mathbf{x}_p + \mathbf{x}_n \):**

#### (a) For \( x + y + z = 4 \):  
   - Convert the equation to matrix form.
   - Solve for the general solution \( \mathbf{x} \) in terms of a particular solution \( \mathbf{x}_p \) and nullspace vectors.

#### (b) For the system:  
\[
\begin{cases} 
x + y + z = 4 \\ 
x - y + z = 4 
\end{cases}
\]
   - Similarly, express this system in matrix form and solve for the general solution.

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### **(3) Suppose the complete solution is:**

\[
\mathbf{x} = \begin{bmatrix} 1 \\ 3 \end{bmatrix} + c \begin{bmatrix} 0 \\ 1 \end{bmatrix},
\]
where \( c \) is a constant. Find the matrix \( A \).

From the given solution form, the nullspace vector \( \mathbf{n} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \) provides insight into \( A \)'s nullspace. Using this and the particular solution, construct \( A \).

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### Do you want me to start solving each part step by step? Let me know how much detail you'd like!

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### Relative Questions:
1. How is the column space related to the rank of a matrix?
2. What is the geometric interpretation of the nullspace?
3. How does the presence of free variables in \( A \mathbf{x} = \mathbf{0} \) affect the nullspace?
4. What are the implications of full rank in \( A \) on the solution space?
5. How do you interpret the particular solution and the nullspace solution geometrically?

**Tip:** Row-reducing the augmented matrix systematically will simplify finding both the column space and nullspace.

1. Describe the column space and nullspace of the given matrix A, and find the complete solution x for Ax = b with b = [4, 3, 5]. 2. Find the complete solution in the form x = xp + xn for the given systems. 3. Given a specific solution form, determine the matrix A.

Explore the solution for Ax = b with b = [4, 3, 5] by analyzing the column space and nullspace of matrix A. Learn to find solutions in the form x = xp + xn using Gaussian elimination and vector space principles. Understand how to reconstruct matrix A from specific solutions.

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