For which value $t$ we get a basis in $\mathbb{R}^{3}$ by these $3$ vectors?

Question

For which $t \in \mathbb{R}$ we have a basis in $\mathbb{R}^{3}$ if the vectors are

$$\begin{pmatrix} 1\\ 2\\ 2t \end{pmatrix}, \begin{pmatrix} 1\\ 2t\\ 0 \end{pmatrix}, \begin{pmatrix} 1\\ t\\ 1+t \end{pmatrix}$$

This is no homework, just another task that could be asked in my linear algebra exam.

I have no idea how to solve such a task. I have tried Gauss but it didn't work (I could get $3$ zeroes in bottom left corner of the matrix but getting $3$ more zeroes in top right seemed impossible).

Then I thought try to insert some value for $t$ such that the determinant $\neq 0$.

So I just took $t = 0$ because the determinant would be $-2$ and thus the vectors would be linearly independent which indicates they are a basis.

But then there would be many solutions... Did I do it correctly anyway? Is the notation correct too ?

Answer 1

I think the easiest way is to use the determinant. Then You can put the polynomial equal to zero.

You should put your vectors in columns of a $3 \times 3$ matrix. If the determinant of a matrix is $0$, then it means that the columns are linearly dependent and they cannot make a basis for $\mathbb{R}^3$. Therefore, you calculate the determinant of the matrix, which is a polynomial with respect to $t$, and put it equal to zero.

What you achieve at the end is

$-2=0$

It means that for no value of $t$ the determinant can be $0$ and the vectors are always linearly independent.