Basis R 2


by Dustinsfl
Tags: basis, linear algebra, vector space
Dustinsfl
Dustinsfl is offline
#1
Feb28-10, 02:06 PM
P: 628
x1= column vector (2, 1)
x2= column vector (4, 3)
x3= column vector (7, -3)

Why must x1, x2, and x3 be linearly dependent?

x1 and x2 span R^2.
The basis are these two columns vectors: (3/2, -1/2), (-2, 1)

Since x1 and x2 form the basis, x3 can be written as a linear combination of these vectors.

Is that it? or correct?
Phys.Org News Partner Science news on Phys.org
Going nuts? Turkey looks to pistachios to heat new eco-city
Space-tested fluid flow concept advances infectious disease diagnoses
SpaceX launches supplies to space station (Update)
LCKurtz
LCKurtz is offline
#2
Feb28-10, 02:17 PM
HW Helper
Thanks
PF Gold
LCKurtz's Avatar
P: 7,198
Quote Quote by Dustinsfl View Post
x1= column vector (2, 1)
x2= column vector (4, 3)
x3= column vector (7, -3)

Why must x1, x2, and x3 be linearly dependent?
How to answer that question depends on what you have learned. What is the dimension of R2?
x1 and x2 span R^2.
The basis are these two columns vectors: (3/2, -1/2), (-2, 1)
There is no such thing as the basis for R2. Any two linearly independent vectors in R2 are a basis.
Since x1 and x2 form the basis, x3 can be written as a linear combination of these vectors.

Is that it? or correct?
You could just demonstrate x3 = cx1 + dx2; that would surely settle it.
Dustinsfl
Dustinsfl is offline
#3
Feb28-10, 02:32 PM
P: 628
New question:
x1=(3, -2, 4)
x2=(3, -1, 4)
x3=(-6, 4, -8)

What is the dimension of span (x1, x2, and x3)

The book says 1; however, shouldn't the dimension be 3? I see that these 3 vectors are all the same times a constant but there are coordinates.

LCKurtz
LCKurtz is offline
#4
Feb28-10, 02:58 PM
HW Helper
Thanks
PF Gold
LCKurtz's Avatar
P: 7,198

Basis R 2


Quote Quote by Dustinsfl View Post
New question:
x1=(3, -2, 4)
x2=(3, -1, 4)
x3=(-6, 4, -8)

What is the dimension of span (x1, x2, and x3)

The book says 1; however, shouldn't the dimension be 3? I see that these 3 vectors are all the same times a constant but there are coordinates.
If they are supposed to be a constant times each other you have mistyped something. But assuming that, what is the definition of dimension that you are using? You have to apply that.


Register to reply

Related Discussions
Basis for the set of all cts fns? Linear & Abstract Algebra 15
RGB basis? General Math 6
matrix connecting Sz diagonal basis to Sx diag basis Quantum Physics 0
Basis independent and basis dependent formulation of QM Advanced Physics Homework 0
free basis and basis? Calculus & Beyond Homework 7