# Basis R 2

by Dustinsfl
Tags: basis, linear algebra, vector space
 P: 588 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?
HW Helper
Thanks
PF Gold
P: 6,994
 Quote by Dustinsfl 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.
 P: 588 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.
HW Helper
Thanks
PF Gold
P: 6,994

## Basis R 2

 Quote by Dustinsfl 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.

 Related Discussions Linear & Abstract Algebra 15 General Math 6 Quantum Physics 0 Advanced Physics Homework 0 Calculus & Beyond Homework 7