Question

Linear Algebra: What condition must be placed on a, b, c so that the system has at least one solution. (see attached)

Answer 1

\[\text{$[a, b, c]$ must be in the image of the transformation}\]\[\left(\begin{matrix}a \\ b \\ c\end{matrix}\right) \in \left\{ c_1\left(\begin{matrix}1 \\ 2 \\ 1\end{matrix}\right) + c_2\left(\begin{matrix}2 \\ 6 \\ -2\end{matrix}\right) \| c_1, c_2 \in \mathrm{F}\right\}\]

Answer 2

\[det \left[ \begin {array}{ccc} 1&2&-3\\ 2&6&-11
\\ 1&-2&7\end {array} \right] = 0\]\[rank \left[ \begin {array}{ccc} 1&2&-3\\ 2&6&-11
\\ 1&-2&7\end {array} \right] =2\]

Answer 3

alchemist:sorry my mistake (i'll remove the erroneous post).
nisha:he's right his answer stems fromt the fact that if |A| = 0 then to make the system consistant \[A_{x_1}\]... also have to be zero and theya re zero when 2 of the columns or rows are equal

Answer 4

alch:could you elaborate on your condition for the solution....especially touching on its completeness (i.e. it containing all possible solutions) ....

Answer 5

What about the conditions on a, b, and c?