0 votes 0 votes Question: How NullSpace of the matrix A and the uniqueness of the solution of Ax=b are related ?? Linear Algebra self-doubt linear-algebra + – lalitver10 asked Dec 24, 2022 lalitver10 295 views answer comment Share Follow See 1 comment See all 1 1 comment reply ankitgupta.1729 commented Dec 24, 2022 reply Follow Share $1)$ If $K$ be the solution set of a system of linear equations $Ax=b$ and $K_H$ be the solution set of the corresponding homogeneous system $Ax = 0$ then for any solution $s$ to $Ax=b$: $$K= \{s\} + K_H = \{s + k : k \in K_H\}$$ (It can be proved easily) $2)$ Let $Ax = b$ be a system of $n$ linear equations in $n$ unknowns. If $A$ is invertible, then the system has exactly one solution, $A^{−1}b.$ and Conversely, if the system has exactly one solution, then A is invertible. Now, answer to your question is based on the above two facts: Suppose that system has only one solution $s$ and let $K_H$ be the solution set for corresponding homogeneous system $Ax=0$. Now, according to $(1),$ $\{s\} = \{s\} + K_H$ and so, $K_H = \{0\}$ and Thus null space, $N(L_A) = \{0\}$ and hence $A$ is invertible. 1 votes 1 votes Please log in or register to add a comment.