Linear Dependence of x1, x2 and x3 in R^2

[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?

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?

 

[LCKurtz]

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]

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]

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.