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A is a $5 \times 3$ matrix that has column vectors {$v_1, v_2, v_3$}, the only solution to $Ax = 0$ is $x= 0$, and the set of vectors {$v_1, v_2, v_3, b $} is linearly dependent. Is the system $Ax = b$ consistent, inconsistent or could be either one?

So far, I understand that if a $Ax = 0 $ has only the trivial solution ($x = 0$), then its columns are linearly independent. This means that the column vectors of A are linearly independent. Since the set of vectors {$v_1, v_2, v_3, b $} is linearly dependent, this leads me to believe that b is a combination of the columns of A, but I'm not sure how that affects the whether or not the system is consistent.

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For $x=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}$ you can compute $$Ax=x_1v_1+x_2v_3+x_3v_3.$$ so, if $b=\alpha v_1+\beta v_2+\gamma v_3$, can you find $x$ so that $Ax=b$?

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Hint: For the equation $Ax=b$ to have any solution at all, $b$ must be an element of the column space of $A$.

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Your reasoning about $b$ being a Linear Combination of the columns of $A$ is correct. Just realize now that $b=(\alpha_1\cdot v_1)+(\alpha_2\cdot v_2)+(\alpha_3\cdot v_3)\Rightarrow Ax=b \ where \ x=\{ \ \alpha_1,\alpha_2,\alpha_3\}$

Thus, the system has at least one solution and must be consistent.

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