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.
Best Answer
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 $Ax=b$?