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Basis R 2

by Dustinsfl
Tags: basis, linear algebra, vector space
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Dustinsfl
#1
Feb28-10, 02:06 PM
P: 629
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?
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LCKurtz
#2
Feb28-10, 02:17 PM
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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
#3
Feb28-10, 02:32 PM
P: 629
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
#4
Feb28-10, 02:58 PM
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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.


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