Check if three row vectors are linearly dependent or independent

Question

I want to check if three row vectors are linearly dependent or independent. I have the following row vectors: $u = \left[\begin{array}{r}1 & 3 & -1 & 2\end{array}\right], v=\left[\begin{array}{r}2 & -2 & 5 & 1\end{array}\right], w = \left[\begin{array}{r}1 & -1 & 2 & -1 \end{array}\right]$.

Now, if they are linearly independent it should mean that I would be able to find some non-zero scalars $\lambda_1, \lambda_2, \lambda_3$: such that I am able to satisfy the following equation:

$$\lambda_1u+\lambda_2v+\lambda_3w=0$$

However, I am not really able to understand how to proceed from here:

$$\lambda_1\left[\begin{array}{r}1 & 3 & -1 & 2\end{array}\right]+\lambda_2\left[{\begin{array}{r}2 & -2 & 5 & 1\end{array}}\right]+\lambda_3\left[{\begin{array}{r}1 & -1 & 2 & -1\end{array}}\right] = 0$$

Please note that my textbook has not yet talked about rank or determinants. Any hints on how to solve this without those? Should I solve a linear system? Should I build a matrix using these row vectors as rows?

Answer 1

You are always able to find such scalars: just take $\lambda_1=\lambda_2=\lambda_3=0$. The question is: is there some other solution? If there is, the vectors are linearly dependent. Otherwise, they are linearly independent.

So, solve the system$$\left\{\begin{array}{l}\lambda_1+2\lambda_2+\lambda_3=0\\3\lambda_1-2\lambda_2-\lambda_3=0\\-\lambda_1+5\lambda_2+2\lambda_3=0\\2\lambda_1+\lambda_2-\lambda_3=0\end{array}\right.$$Is there some solution besides the null solution or not?

Answer 2

A much quicker and easier approach is to form a matrix from these vectors and see if it has full rank. In this case, if these vectors are linearly independent, the rank has to be $3$. It turns out that the rank is indeed $3$, so the given vectors are linearly independent.