Solutions to T(x)=b

Proved by: Not Applicable
References: Kernel, Image of Linear Transformation
Justifications: Not Applicable

Specializations: Not Applicable
Generalizations: Not Applicable

Solutions to $T(\vec{x})=\vec{b}$

Let be a linear transformation. The system of linear equations has

1. at most one solution for every iff .
2. at least one solution for every iff .

Proof.

1. If the kernel of is not , then there is more than one solution to . (Of course, one solution is ).
   In the other direction, if there exists a for which has more than one solution, i.e.,
   
   
   So is a nonzero element of the kernel, so .
2. Saying that is exactly saying that is onto.