Basis Vectors, Linear Combinations, Span and Linear Independence/Dependence
Linear Combination
Firstly a set of basis vectors is a set of vectors that we can scale and add to each other to reach every point in space... For example, we usually define the basis vectors of 2D space as (i.e. one step in the x-direction) and (i.e. one step in the y-direction) because we can use these vectors to reach every point in 2D space but we could have equally used or etc.
If we add multiples of vectors to each other this is called a linear combination
Geometrically, when we use a linear combination we are just joining the tips of vectors together to form a resultant vector and then seeing what happens when we scale each individual vector.
Span
A span is the set of all resultant vectors that we can get by using a linear combination of the set of vectors that we have. The space which these vectors span is called the vector space.
Usually, the span will be all of space given by the dimensions of the vectors. For example, if we have two-dimensional real vectors then the span will usually be all of 2D space which we call . If we have three-dimensional real vectors then the span will usually be all of 3D space which we call . If we have n-dimensional real vectors then the span will usually be all of nD space which we call .
If this is the case, we call our set of vectors linearly independent
Linear Dependent Example
Linear Dependent Vectors
If however, our set of vectors do not span all of then we say they are linearly dependent
This is will happen if our vectors are lying on the same line (or same plane in 3D) and so one will just be a scaled version of the other. Hence, when we come to write a linear combination, at least one of the vectors will be redundant (since the idea is to scale vectors and then add them to each other to achieve new vectors...but clearly this will just get new vectors that we could have got by scaling just one of them).
Hence we say that a set of vectors are linearly dependent if we can write one of them as a linear combination of the others.
e.g. if and then clearly they are linearly dependent because they lie on the same line but also so we can write as a linear combination of