What real values of $c$ makes these two vectors linearly independent?

I had a question from my homework Friday night which stumped me.

Find all real values of $$c$$ such that the vectors $$\begin{bmatrix}2\\c \\0 \end{bmatrix}, \begin{bmatrix}c\\c+3 \\0 \end{bmatrix}$$

are linearly independent.

I got stumped by this question; after trying to row reduce, I got

$$\begin{bmatrix} 2c &c^2 & |0 \\ 0& -\frac{1}{2}c^2+c+3& |0 \\ 0& 0& |0 \end{bmatrix}$$

I wasn't sure what to do at this point. I see the two quadratics; I'm just unsure of what to do and why.

• Hint: two vectors $a,b \in \mathbb{R}^3$ are linearly independent if $a \times b \neq 0$ (cross product). Feb 21, 2021 at 21:02
• The vectors are effectively $2$-vectors; just evaluate the two-by-two determinant. Feb 21, 2021 at 21:09

Suppose the vectors are linearly dependent ie. $$\begin{bmatrix}2\\c \\0 \end{bmatrix}=\lambda \begin{bmatrix}c\\c+3 \\0 \end{bmatrix}$$ with $$\lambda \neq 0$$. So $$c^2 = 2(c+3)$$ ie. $$c=1 \pm \sqrt{7}$$. And if $$c=1 \pm \sqrt{7}$$ the vectors are linearly dependent. So the vectors are linealy independent iff $$c \in \Bbb R - \{1\pm \sqrt{7}\}$$.
You can also use row-reduction: let $$A=\begin{bmatrix}1 & c/2 \\ 0 & c+3-c^2/2 \end{bmatrix},$$ and use the fact that the vectors are linearly independent iff $$\ker A = \{0\}$$.