# Determining whether a system is consistent or inconsistent from the linear dependence of its columns

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.

## 3 Answers

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 $$Ax=b$$?

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

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.