# What is significance of linear independence?

If there is vectors, $$v_1,v_2,...$$ .

They are linearly independent if $$c_1v_1+c_2v_2+... = 0$$ with $$c_1=c_2=...=0$$.

• if $$v_1,v_2,...$$ are linerly independent and so what?

• what will the linear independence of vectors lead to? Base?

• Do linear independence has any connection to dot product or orthogonality?

• what will happen if they are not linearly independent?

Linearly independence means $$c_i=0$$ is the only way to achieve $$c_iv_i=0$$. It follows you can't write any one vector as a linear combination of the others, so the space they span together can't be spanned by any proper subset of them. In an inner product space, you can make an orthogonal basis from independent vectors, but they're not guaranteed to be one.