Determine whether the vectors $(1,3)^T$,$(1,1)^T$,and $(0,1)^T$ are linearly independent or not.

Question

Determine whether the vectors $(1,3)^T$,$(1,1)^T$,and $(0,1)^T$ are linearly independent or not.

My attempt

Suppose that $$a(1,3)^T+b(1,1)^T+c(0,1)^T=(0,0)^T$$ $$(a,3a)^T+(b,b)^T+(0,c)^T=(0,0)^T$$ $$(a+b,3a+b+c)^T=(0,0)^T$$ $$a+b=0,3a+b+c=0$$

I know that $a=b=c=0$ then only the vectors will be linearly independent.

But,I am stuck here. Anyone please explain how to solve after this.

Answer 1

They are not independed as the dim of R^2 is only 2.....your system has not only the 0 solution....

Answer 2

You could take $a=-b$, which gives $c=2b$. For instance $a=-1,b=1$ and $c=2$ is a nontrivial solution.

The dimension argument: $3$ vectors in a $2$-dimensional space cannot be linearly independent...