Consider a linear system with 7 equations for 8 unknowns. Circle all possible types of solution sets that could result:
(a) The system has no solutions.
(b) The system has a unique solution.
(c) The system has exactly 8 distinct solutions.
(d) The system has a one-parameter family of solutions.
(e) The system has a two-parameter family of solutions.
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Consider the matrix representation of the system.
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First write the linear system in the form of where is the coefficient matrix with rows and columns, is a vector representing the unknowns while is a vector representing the right-hand side. Now we perform Gaussian Elimination to the augmented matrix to get its reduced row echelon form. We call the reduced row echelon form .
If there is a zero row in but the entry of on this row is non-zero, then there is no solution. So (a) is possible.
The system has a unique solution if the corresponding homogeneous system has zero solution. Note that the matrix has columns and rows, its column spaces are linear dependent. This implies that there is a nonzero vector in the null space, so (b) is wrong.
(c) is not possible. If we assume for the sake of contradiction that the linear system has exactly eight solutions , then also solves the linear system, yielding a contradiction.
The matrix has columns and rows, the maximal possible rank of matrix is If has rank , that is, no zero rows in , it is possible for the linear system to have one parameter family of solutions.
If there is one zero row in the reduced row echelon form , then the matrix has rank . In this case, it is possible for the system to have two parameter family of solutions.
Thus, the possible results are .
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