Dependent and independent Vectors

Question

I am reading the Linear Algebra. I have doubt on dependent or Independence of vectors.

What I understand is following

1.Vectors that doesn't span a space are independent.

2.If they span a space that means they have some combination in their own space and I will get the less Vector.Can I say they are dependent ?

Am I going into the right direction while understanding the linear algebra. Are these points correct. Please rectify if I am not correct.

I am just novice in Linear algebra.

Answer 1

No it is not correct.

The definition is that $n$ vectors $v_i\neq 0$ are linearly independent when

$$\sum a_i v_i=0 \iff a_i=0$$

As a consequence we have that

1. $n$ vectors that do not span an $n$ dimensional space are not independent

2. if they span an $n$ dimensional space that means they are linearly independent

Answer 2

Every nonempty set of vectors will span some space. I think it might help you to focus first on understanding the idea of linear combinations. All three of the ideas you mentioned depend on that one.

The span of your set of vectors is the set of all possible linear combinations of those vectors.

If any one vector in set can be written as a linear combination of the other vectors then the set is dependent. Otherwise it is independent.

Answer 3

I don't think there is a good way to "understand linear dependence in terms of span", if you consider "span" as a black box operation.

You could say that a set of vectors is linearly independent if its span has the same dimension as the number of vectors you start with -- but the concept of "dimension" itself depends on knowing what linear independence means, so that is not really progress.

Perhaps you could base an understanding of the kind you want to say that linear independence is about how the set of vectors generate its span. By definition each element of the span is a linear combination -- but if every vector in the span has only one linear combination that produces it, then the set is linearly independent.

It turns out that you don't need to check every vector in the span because it can be proved that an arbitrary vector in the span has as many linear combinations that produce it as the zero vector has. So it's enough to ask whether there is more than one linear combination for the zero vector. And since the trivial linear combination always produces the zero vector, this is the same as asking whether there's any linear combination other than the trivial one that produces the zero vector.