That is the definition of linear independence.
#Linear independent vectors
grim fern
drowsy ivy
Yeah but why is it
grim fern
What do you mean "why"? It is the definition.
A collection of vectors are linearly dependent if there is a nontrivial linear combination of them that is equal to the zero vector.
drowsy ivy
And why is this the definition
grim fern
I don't understand the question.
sterile arch
Probably asking why we defined it like that
grim fern
Oh.
drowsy ivy
I don't expect the man who came up with this definition he had nothing in mind and decided on this thing to be "linear independence"
grim fern
Well, that's because we are usually interested whether n vectors make up a basis of ℝ^n (or the n-dimensional space they are defined in, anyway).
And that only happens when they are linearly independent.
sterile arch
Right and the definition shows that vectors are linearly independent
Because let's say a, b and c are not all zero right
And the sum is still equal to 0
Then that must mean that one vector is a scalar multiple of another
And hence linearly dependent
grim fern
Then that must mean that one vector is a scalar multiple of another
Well, that doesn't necessarly mean that.
But it might happen.
grim fern
Oh, or did you mean the nontrivial linear combination of them is equal to zero?
Then yes.
sterile arch
Yes sorry my English isn't all that great at times haha
drowsy ivy
Oh, or did you mean the nontrivial linear combination of them is equal to zero?
What is a trivial and nontrivial linear combination?
trivial like all the coefficients are 1?
Or what
grim fern
What is a trivial and nontrivial linear combination?
A linear combination is trivial if all of its coefficients are 0, and nontrivial otherwise.