Given two vectors $(1,2,8),(0,1,9)$ find a 3rd vector that is linearly independent from these two vectors.
I sort of have an idea how to go about solving the problem but I'm not 100% sure. I'm know we just want to find a vector that can't be written as the sum of the two given vectors but how exactly do I go about finding one such vector?
Got it now many thanks to all the helpers some brilliant explanations.
Best Answer
I like your idea about finding a vector that can't be written as a sum of the two vectors above. Let's take a look at what that would look like.
Every possible sum of these two vectors can be expressed as $c_{1}(1,2,8) + c_{2}(0,1,9)$ for some $c_{1}, c_{2}$ in $\Bbb R$. So, all possible sums can be expressed in the form $(c_{1}, 2c_{1} + c_{2}, 8c_{1} + 9c_{2})$.
We want to come up with a third vector $(v_{1}, v_{2}, v_{3})$ that can't be expressed in the above form. Whatever $c_{1}$ and $c_{2}$ you pick for the linear combination above, we need that the third component $v_{3}$ is exactly $8c_{1} + 9c_{2}$. Let's pick a vector whose third component is different from this (i.e., pick $c_{1}$ and $c_{2}$, and fill in the first two components of $(c_{1}, 2c_{1} + c_{2}, 8c_{1} + 9c_{2})$, but make the third component different from this).
So, even though you can pick $c_{1}$ and $c_{2}$ to be anything, I will pick $c_{1} = c_{2} = 1$. Then the vector I will construct will be:
$(1c_{1}, 2c_{1} + 1c_{2}, 11) = (1, 2 + 1, 11) = (1, 3, 11)$
Notice that I made the third component different from $8c_{1} + 9c_{2} = 8 + 9 = 17$. Then this new vector can't be written as a linear combination of the previous two vectors, because that's how we constructed it.