Suppose a matrix A is nxn, and that v1,v2 are in R^n...

Question

Sorry, this is my first time posting here, so if my question is worded incorrectly, please let me know. Anyway, I'm studying for an exam coming up, and this is one of the questions that I'm trying to work through.

Suppose a matrix, A, is nxn and that vectors v1, v2 are within R^n. Suppose Av1 = 4v1 and that Av2 = 7v2. Show that {v1,v2} is a linearly independent set of vectors.

Answer 1

Suppose they are linearly dependent, then there are constants $c_1,c_2$ so that

$$ c_1v_1 + c_2v_2 = 0 $$

Actually, because you are only dealing with two vectors, you can say this differently: there is $c \neq 0$ so that

$$ v_1 = cv_2 $$

What does this tell you if you compare $Av_1$ and $Av_2$ in two different ways: using what you know about eigenvalues, and using the fact that $v_1 = cv_2$.

Answer 2

Assume $v_1,v_2$ are not independent, then each one is the scalar multiple of the other, say $v_1=\alpha v_2$, where $\alpha\neq 0$. This means

$$Av_2=7v_2\Rightarrow Av_1=7v_1$$

contradicts the assumption $Av_1=4v_1$