Linear Dependence/Independence Help

[vg19]

Hi,

I am struggling with the following question and don't know where to start. I would appreciate any help.

Thanks

Determine whether the following sets of vectors form bases for two dimensional space. If a set forms a basis, determine the coordinates of v = (8,7) relative to this base

a) v1 = (1,2) v2 = (3,5)
b) v1 = (3,5), v2 = (6,10)

Thanks!

 

[Tide]

What exactly have you tried so far and what is your thinking?

 

[vg19]

Hi,

To be honest, I havnt done anything with this question because I really do not know where to start. Actually, I do not really understand what this question is asking me to do in the first place. What exactly does "form bases for two-dimensional space" mean? And when they said "relative to this base" what does that mean?

Thanks

 

[Tide]

I'm not sure where to begin.

In ordinary "Cartesian" space you can express any vector as a sum of the two basis vectors (1, 0) and (0, 1), i.e. the unit vectors along the x and y axes respectively. Those vectors happen to be orthogonal and, for most purposes, are quite handy and useful. However, they may not be the most convenient basis vectors in certain situations such as in analyzing certain crystal lattices where you may want basis vectors more closely resembling the structure of the lattice.

In order to accomplish that you have one overriding constraint: the basis vectors you choose must not be collinear. Your example (b) uses (3, 5) and (6, 10) which are obviously collinear since multiplying the first by 2 gives the second. Your first example with (1, 2) and (3, 5) works fine! Any vector in your 2-D space can be written as a linear combination of those two basis vectors.

All you need to do is to find the value of, say, A and B such that A*(1, 2) + B*(3, 5) gives the vector in question. In your case, find A and B such that A*(1, 2) + B*(3, 5) = (8, 7).

Didn't they cover any of this in the course you enrolled in?

 

[HallsofIvy]

If you are trying to prove that a given set of vectors is a basis, then a reasonable START would be to look up the definition of "basis"!