I have an exercise that asks me to solve for the basic solution of a homogenous linear system and express the general solution as a linear combination of the basic solution. It gives me the matrix of the coefficients, I though I was suposed to simply solve the linear equations system, but it always give me only the trivial solution. I could not find the definition for basic solution and general solution ( I know preety well what a linear combination is though ). Thank you in advance !
[Math] the basic and the general solution for a homogeneous linear system
linear algebramatrices
Best Answer
If $A$ is an $(m\times n)$-matrix the solutions of the homogeneous system $$Ax=0\tag{1}$$ form a subspace $V$ of ${\mathbb R}^n$. The dimension $d$ of this subspace is given by $d=n-{\rm rank}(A)$.
The solution space $V$ has various bases $(v_1,v_2,\ldots, v_d)$ with $v_i\in{\mathbb R}^n$, all of them containing exactly $d$ vectors. If in a concrete example you have found that (1) only has the trivial solution then ${\rm dim}(V)=d=0$, and any basis of $V$ is the empty set.
Solving $(1)$ means establishing a parametric representation of the set $V$. In essence this amounts to producing a basis $(v_1,v_2,\ldots, v_d)$ in explicit numerical form and then declaring that $$V=\bigl\{\lambda_1 v_1+\ldots+\lambda_d v_d\>\bigm|\>\lambda_i\in{\mathbb R} \ (1\leq i\leq d)\bigr\}\ .\tag{2}$$The Gaussian elimination procedure results in a statement of the form "$x_1$, $\ldots$, $x_d$ arbitrary, and $x_k=\sum_{i=1}^d b_{ki}x_i$ $\>(d+1\leq k\leq n)$. But this is $(2)$ in disguise.