What is the span of {(1; 1; 0),(0; 0; 2)}?

[negation]

Homework Statement

What is the span of {(1; 1; 0),(0; 0; 2)}?

The Attempt at a Solution

So the span is λ₁(1; 1; 0) + λ₂(0; 0; 2)

But how should I express my answer?
The given answer on the sheet is as below:
The span is {(λ,λ,β)|λ,βεR}

 

[Mark44]

Homework Statement

What is the span of {(1; 1; 0),(0; 0; 2)}?

The Attempt at a Solution

So the span is λ₁(1; 1; 0) + λ₂(0; 0; 2)

The span is all linear combinations of <1, 1, 0> and <0, 0, 2>, which is the same span as all linear combinations of <1, 1, 0> and <0, 0, 1>.

Using the same letters as in the answer, the span is all vectors of the form λ<1, 1, 0> + β<0, 0, 1> = <λ, λ, 0> + <0, 0, β>, or more simply, <λ, λ, β>, with λ and β arbitrary reals.

But how should I express my answer?
The given answer on the sheet is as below:
The span is {(λ,λ,β)|λ,βεR}

 

[negation]

The span is all linear combinations of <1, 1, 0> and <0, 0, 2>, which is the same span as all linear combinations of <1, 1, 0> and <0, 0, 1>.

Using the same letters as in the answer, the span is all vectors of the form λ<1, 1, 0> + β<0, 0, 1> = <λ, λ, 0> + <0, 0, β>, or more simply, <λ, λ, β>, with λ and β arbitrary reals.

In red, where did <0,0,1> came from? 2<0,0,1> = <0,0,2>?

Edit: Alright, let's not complicate things since my intention was only to know why the answer was expressed in the way it was.
Thanks

 

[Mark44]

In red, where did <0,0,1> came from? 2<0,0,1> = <0,0,2>?

Yes, <0, 0, 2> is a scalar multiple of <0, 0, 1>. You could just as well have picked <0, 0, 17>.

Edit: Alright, let's not complicate things since my intention was only to know why the answer was expressed in the way it was.
Thanks