# Proving linear independence

Let $A$ be an $n \times n$ matrix and suppose $v_1, v_2, v_3 \in \mathbb{R}^n$ are nonzero vectors that satisfy:

$$Av_1 = v_1 \\ Av_2 = 2v_2 \\ Av_3 = 3v_3$$

Prove that $\{v_1, v_2, v_3\}$ is linearly independent. (Hint: Start by showing that $\{v_1, v_2\}$ must be linearly independent.) I know if $A$ is non-singular,it is easy. But if $A$ is singular, I have no idea to get that.

• Do you know about eigenvectors and eigenvalues? – EuYu Apr 12 '13 at 4:26
• Yes,but i do not think this question requires this technique – Yunhui Shi Apr 12 '13 at 4:30
• if matrix is singular,it means that it's columns or rows are also linear dependent,maybe you should use this fact – dato datuashvili Apr 12 '13 at 5:01
• The eigenvectors corresponding to distinct eigenvalues are always linearly independent – Shahab Apr 12 '13 at 5:36

## 1 Answer

HINTS

1. Suppose $v_1$ and $v_2$ are linearly dependent, so that $v_1 = cv_2$.
2. What is $Av_1$? What about $A cv_2$? What about $cAv_2$?
3. Do the same idea for $v_3$ independence.