Linear dependence/independence of vectors

[Yitzi_K]

Could someone please explain this topic to me? My teacher didn't do the best job, and I'm somewhat confused about the whole thing.
Thanks

[f.sharp]

if 2 vectors can be used to represent a 3rd, they are linearly dependent. say 3 vectors that form a triangle, the sum of all 3 equals 0. if not then theyre independent.

oh and if u have 2 vectors, they are almost always independent UNLESS one is the scalar multiple of the other.

[Martoman]

if 2 vectors can be used to represent a 3rd, they are linearly dependent. say 3 vectors that form a triangle, the sum of all 3 equals 0. if not then theyre independent.

oh and if u have 2 vectors, they are almost always independent UNLESS one is the scalar multiple of the other.

Great explanation. The idea of forming a triangle with three vectors is how I see it. With two vectors only if they are parallel they are linearly dependent. A trivial proof of this is that:

ie b is parallel to a, then ie they will always form a linearly dependent set.

[Yitzi_K]

if 2 vectors can be used to represent a 3rd, they are linearly dependent. say 3 vectors that form a triangle, the sum of all 3 equals 0. if not then theyre independent.

oh and if u have 2 vectors, they are almost always independent UNLESS one is the scalar multiple of the other.

When you say that the 3 vectors added together equals 0, is that a + b + c = 0, or ka + pb + qc= 0?

[jasoN-]

its the latter, I should know, got it wrong in a recent sac costing me full marks (i used a + b + c = 0)

To add instead of using ka + pb + qc= 0
just use a = pb + qc, the real coefficients will be different but its more simple to get an answer

[m@tty]

To add instead of using ka + pb + qc= 0
just use a = pb + qc, the real coefficients will be different but its more simple to get an answer

Definitely.


Also worthy of note that three vectors in two dimensions are ALWAYS dependent. Same with 4 in 3 dimension, n+1 vectors in n dimensions.

[Mao]

A quick note on the interpretation:

A set of vectors are linearly independent if no vectors in the set can be represented by a linear combination of other vectors in the same set. That is, if a set of vectors {a, b, c, d} is linearly independent, then a cannot be expressed in terms of b, c and d, and b cannot be expressed in terms of a, c and d, and so on.

Independent sets are often useful to describe a 'span'. A 2D plane can be completely spanned by the two vectors . That is to say, any vectors in the 2D plane can be described by a linear combination of these two.

If you have a set of vectors , this set of vectors is not linearly independent because . That is, spans all linear combinations of . In this sense, is 'redundant'.

[Yitzi_K]

Thanks a lot guys this has been a real help.