# Math Help - linear combination.. help please

1. ## linear combination.. help please

Is vector u = [3 3 2] is a linear combination of vector v1 =[1 0 1] and
vector v2 =[1 1 1] ?

could you please show me how to solve this problem? thank you very much.

2. Hello,
Originally Posted by happystudent
Is vector u = [3 3 2] is a linear combination of vector v1 =[1 0 1] and
vector v2 =[1 1 1] ?

could you please show me how to solve this problem? thank you very much.
This question can be answered using the determinant the three vectors : $u$ is a linear combination of $v_1$ and $v_2$ iff the determinant $D=\left|\begin{smallmatrix} 3&1&1\\3&0&1\\2&1&1\end{smallmatrix}\right|$ equals 0. Do we have $D=0$ ?

Another way of answering this question is solving $u=av_1+bv_2$ for $a,b\in\mathbb{R}$. If this equation has at least one solution then $u$ is a linear combination of $v_1$ and $v_2$. If it has no solution then...

3. In order for vector u = [3 3 2] to be a linear combination of vector v1 =[1 0 1] and vector v2 =[1 1 1], it would necessary u=3[1,1,1]+t[1,0,1]. That is the only way to have 3 in the second position of u. But that means that 3+t=3 and 3+t=2. But that is impossible.