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2 $begingroup$ I'm trying to find the following set (some authors call it "column space"): Given a matrix $A in mathbb{R} ^{m times n}$ , we define: $$R(A)={b in mathbb{R}^{m} | exists space xin mathbb{R} ^n, b=Ax }.$$ Let $A= left[ begin{array}{ccc} 2 & 2 & 0 & 0 \ 3 & 4 & -1 & 2 \ -1 & 1 & -2 & 4 end{array} right]$ . My attempt goes as follows: I'm trying to find the elements of $R(A)$ . Notice that the elements of $R(A)$ are vectors in $mathbb{R} ^3 $ . I want to know how those elements are. If I take a generic vector $b=(b_{1},b_{2},b_{3})^T in mathbb{R}^3$ , then I'm trying to solve the following equation system: $$ left[ begin{array}{ccc} 2 & 2 & 0 & 0 &|& b_{1} \ 3 & 4 & -1 & 2 &|& b_{2}\ -...