## Linear combinations

In the vector space $$\mathbb R^4$$ over the field $$\mathbb R$$ find the linear combination of vectors $$(-5{,}5,1,-1)^T$$, $$(2,-5{,}0,2)^T$$, $$(3{,}2,0,-2)^T$$ a $$(2,-3{,}1,1)^T$$ which does lead to vector $$(-7{,}12,2,-4)^T$$. Is this linear combination unique?
It is possible to find coefficients of of the linear combination with use of a system of equations. We may obtain for example $$(2{,}0,1{,}0)^T$$ as a solution.
The system of equations does not have a unique solution, thus the linear combination is not unique. The general form of the solution is $$(2{,}0,1{,}0)^T+p(-1,-2,-1{,}1)^T$$.