To find a matrix $A$ such that $W = \text{col } A$, meaning the span of the columns of $A$ is the set $W$, we can proceed as follows:

1. Identify the elements in $W$: $W = \{a + b, 2a - b, -a - b, 4a + 2b\}$ where $a$ and $b$ are real numbers, so we need to express each element of $W$ as a linear combination of the variables $a$ and $b$.

2. Rewrite each element in terms of $a$ and $b$:

   • $w_1 = a + b$
   • $w_2 = 2a - b$
   • $w_3 = -a - b$
   • $w_4 = 4a + 2b$

3. Express each element as a vector: Each vector can be written in terms of $a$ and $b$, so we rewrite $W$ as: $W = a \begin{bmatrix} 1 \\ 2 \\ -1 \\ 4 \end{bmatrix} + b \begin{bmatrix} 1 \\ -1 \\ -1 \\ 2 \end{bmatrix}$ This shows that the vectors $\begin{bmatrix} 1 \\ 2 \\ -1 \\ 4 \end{bmatrix}$ and $\begin{bmatrix} 1 \\ -1 \\ -1 \\ 2 \end{bmatrix}$ span $W$.

4. Form the matrix $A$: Place the spanning vectors as columns in the matrix $A$: $A = \begin{bmatrix} 1 & 1 \\ 2 & -1 \\ -1 & -1 \\ 4 & 2 \end{bmatrix}$

Thus, the matrix $A$ that satisfies $W = \text{col } A$ is: $A = \begin{bmatrix} 1 & 1 \\ 2 & -1 \\ -1 & -1 \\ 4 & 2 \end{bmatrix}$

Let me know if you'd like more details on this process or have further questions!

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Here are five related questions to consider:

1. How can we determine if the columns of $A$ are linearly independent?
2. What does the column space of $A$ represent geometrically in $\mathbb{R}^4$?
3. How do we verify if $W$ spans $\mathbb{R}^4$ or a subspace of it?
4. How would this process change if $W$ contained three variables instead of two?
5. How can we find the rank of $A$ and what does it tell us about the space $W$?

Tip: When expressing vectors as column spaces, always check if they form a basis by confirming they are linearly independent.