In G. Strangs's Introduction to Linear Algebra, 5th, 2.1B of the sample exercises gives equations:
$${x + 3y + 5z = 4}$$ $${x + 2y - 3z = 5}$$ $${2x + 5y + 2z = 8}$$
One of the questions asks to take the dot product of each column of matrix A of coefficients associated with the system of equations above with vector $${y=(1,1,-1)}$$ as well as the dot product of $${y\cdot b}$$ where ${b = (4, 5, 8)}$ and based on the results show that no combination of columns equals ${b}$.
The solution presented shows that each dot product ${Col1\cdot y}$, ${Col2\cdot y}$, and ${Col3\cdot y}$ is 0 while ${y\cdot b = 1}$ which implies ${0 = 1}$ therefore no solution exists.
I do not understand what this process is and why the dot product of the columns with ${y}$ and that of ${b}$ with ${y}$ need to be equal otherwise there are no solutions. Could someone explain this to me?
Let $c_{1},c_{2},c_{3}$ be the column vectors. Suppose the system has a solution $(x_{0},y_{0},z_{0})$. Then $x_{0}c_{1}+y_{0}c_{2}+z_{0}c_{3}=b$. So $y\cdot (x_{0}c_{1}+y_{0}c_{2}+z_{0}c_{3})=y\cdot b$. Hence $x_{0}(y\cdot c_{1})+y_{0}(y\cdot c_{2})+z_{0}(y\cdot c_{3})=y\cdot b$. So $0=1$, contradiction.