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

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In summary, the span of the given vectors is all linear combinations of <1, 1, 0> and <0, 0, 2>, which can also be expressed as all linear combinations of <1, 1, 0> and <0, 0, 1>. Using the same letters as in the given answer, the span can be written as <λ, λ, β>, with λ and β arbitrary reals. This is because <0, 0, 2> is a scalar multiple of <0, 0, 1>.
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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}
 
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negation said:

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.
negation said:
But how should I express my answer?
The given answer on the sheet is as below:
The span is {( λ,λ,β )|λ,βεR}
 
  • #3
Mark44 said:
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
 
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  • #4
negation said:
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>.
negation said:
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
 
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FAQ: What is the span of {(1; 1; 0),(0; 0; 2)}?

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

The span of a set of vectors is the set of all possible linear combinations of those vectors. In this case, the span of {(1; 1; 0),(0; 0; 2)} would be all possible combinations of the form a(1; 1; 0) + b(0; 0; 2), where a and b are any real numbers.

How do you determine the span of a set of vectors?

To determine the span of a set of vectors, you can use the process of Gaussian elimination to find the linearly independent vectors within the set. Then, the span will be all possible linear combinations of those independent vectors.

Is the span of a set of vectors unique?

No, the span of a set of vectors is not unique. The span can vary depending on the number and combination of linearly independent vectors within the set.

What is the relationship between the span of a set of vectors and the dimension of the vector space?

The span of a set of vectors can be used to determine the dimension of the vector space. The dimension will be equal to the number of linearly independent vectors in the set, which will also be the number of vectors needed to span the entire vector space.

Can the span of a set of vectors be larger than the original set?

Yes, the span of a set of vectors can be larger than the original set. This is because the span includes all possible linear combinations of the vectors, which can result in additional vectors that were not in the original set.

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