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I just want to confirm these two questions. Thanks in advance.
(1) Describe all solutions of Ax = 0 in parametric vector form, where A is row equivalent to the given matrix.
[tex]\left(\begin{array}{uvwxyz}1 & 5 & 2 & -6 & 9 & 0 \\0 & 0 & 1 & -7 & 4 & -8\\0 & 0 & 0 & 0 & 0 & 1\\0 & 0 & 0 & 0 & 0 & 0\end{array}\right)[/tex]
There are no solutions because row 3 and 4 contradict each other. Row 3 implies no solution.
(2) Suppose A is a 3x3 matrix and y is a vector in R^{3} such that the equation Ax = y does not have a solution. Does there exist a vector z in R^{3} such that the equation Ax = z has a unique solution?
I said no because if the vector y does not have a solution in R^{3}, then this implies the last row of the row reduced matrix has coefficents that are all zero. Therefore, it either has no solution or an infinite number of solutions.
(1) Describe all solutions of Ax = 0 in parametric vector form, where A is row equivalent to the given matrix.
[tex]\left(\begin{array}{uvwxyz}1 & 5 & 2 & -6 & 9 & 0 \\0 & 0 & 1 & -7 & 4 & -8\\0 & 0 & 0 & 0 & 0 & 1\\0 & 0 & 0 & 0 & 0 & 0\end{array}\right)[/tex]
There are no solutions because row 3 and 4 contradict each other. Row 3 implies no solution.
(2) Suppose A is a 3x3 matrix and y is a vector in R^{3} such that the equation Ax = y does not have a solution. Does there exist a vector z in R^{3} such that the equation Ax = z has a unique solution?
I said no because if the vector y does not have a solution in R^{3}, then this implies the last row of the row reduced matrix has coefficents that are all zero. Therefore, it either has no solution or an infinite number of solutions.