Notation: (A|b) denotes the equation system $Ax=b$
True or false?
Let $K$ be a field, $n\in \mathbb{N}$. Furthermore let $A \in M(n \times n, K)$ a $n \times n$ matrix over $K$ and $b \in K^n$.
a) If the equation system $(A | b)$ has exactly one solution, then for all $c \in K^n$ the equation system $(A | c)$ has exactly one solution.
b) If the equation system $(A | b)$ has no solution, then there exists a $c \in K^n$ such that the equation system $(A | c)$ has more than one solution.
c) If the equation system $(A | b)$ has more than one solution, then the set of all $ c \in K^n$ such that $(A | c)$ has more than one solution forms a true subspace of $K^n$.
d) Let $K$ be a finite field with more than two elements. If the equation system $(A | b)$ has no solution, then the cardinality of the set $M=\{c ∈ K^n |$ the equation system $(A|c)$ has no solution $\}$ is bigger than the cardinality of its complement, $K^n \setminus M$.
My solutions:
a) True. If $(A | b)$ has exactly one solution, then $(A | 0)$ only has got the trivial solution $x=0$. Thus the nullspace is trivial, $A$ is injective and surjective. Thus for every $c \in K^n$ there exists exactly one solution for $(A | c)$.
b) True. If $(A | b)$ has no solution, there exist linearly dependant rows in $A$, thus the rank of $A$ is not full and there exists a non-trivial solution for $(A | 0)$. Given a particular solution for $(A | c)$, the sum of the particular solution and the non-trivial solution for $(A | 0)$ also are a solution for $(A | c)$.
c) True. If $(A | b)$ has more than one solution, then $(A | 0)$ has got the trivial solution $x=0$ and another non-trivial one.
If $(A | b)$ and $(A | b')$ both have more than one solution, then $(A | b+b')$ has got more than one solution, namely the sum of the particular solutions and the trivial solutions.
If $(A | b)$ has got more than one solution, then $(A | \alpha b)$ for $\alpha \in K$ has got more than one solution, namely multiples of $\alpha$ and the particular and the trivial solutions$.
I am not sure about those solutions, especially about b). I also do not know how to tackle d). Any corrections and hints are welcome.
