Determining Basis and Coordinates in Two-Dimensional Space

[winston2020]

Question: 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).


On the first part of the question, I'm a little foggy on how I go about doing it.. I think I have to figure out if they're collinear right? And if they're not, then they can be used to define any other vector in two-dimensional space... is that right?

And so, if that's the case (I believe that they are not collinear), then how do I determine the coordinates of V = (8, 7)? Is it simply a matter of determining the end point of V relative to the base of V1, and V2 with the tails together?

 

[DeadWolfe]

In this case the problem is indeed whether or not they are collinear, but more generally the problem is to figure out whether they are independent. As for finding the coordinates of V relative to that basis, what do coordinates mean? The coordinates are two numbers a and b such that

V=aV1+bV2

But if you write this out, it is just a system of two equations in two unknowns, which you should be able to solve.

 

[winston2020]

In this case the problem is indeed whether or not they are collinear, but more generally the problem is to figure out whether they are independent. As for finding the coordinates of V relative to that basis, what do coordinates mean? The coordinates are two numbers a and b such that

V=aV1+bV2

But if you write this out, it is just a system of two equations in two unknowns, which you should be able to solve.

Thank you. Given what you said, this is what I did:

V = aV1 + bV2
(8, 7) = a(1, 2) + b(3, 5)

Therefore:
8 = a + 3b
7 = 2a + 5b

After solving: a = -(19 / 5), and b = -(38 / 25).

The answer in the book simply says "Yes. (-19, 9)" Can anyone tell me what I'm missing, what I've done wrong here (maybe I just solved a, and b wrong...)?

 

[Vid]

You solved the system wrong. Try substituting a = 8 - 3b into the second equation.

 

[winston2020]

You solved the system wrong. Try substituting a = 8 - 3b into the second equation.

Thanks. The first time I tried substituting b = (7-2(-19/5))/5 into a = 8 - 3b... I just screwed up the fractions. It's all good now though. Thanks everyone