I don't have an idea on how to prove it. What should be my approach?
closed as off-topic by Namaste, Leucippus, Shailesh, Jyrki Lahtonen Apr 9 '17 at 5:22
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Namaste, Leucippus, Shailesh, Jyrki Lahtonen
Here's an idea:
$$a_1 v_1 + a_2v_2 + a_3v_3 = 0$$
Then apply $A$ on both sides to get:
$$A (a_1 v_1 + a_2v_2 + a_3v_3) = A(0)$$
Then by linearity of the matrix operation we have:
$$a_1 Av_1 + a_2Av_2 + a_3Av_3 = 0$$
But since $Av_1, Av_2, Av_3$ are linearly independent, we conclude that:
$$a_1 = a_2 = a_3 = 0 $$
Hint : Suppose that $c_1v_1+c_2v_2+c_3v_3=0$ and apply $A$ on both side.