Let v1→=[23] and v1→=[46] what is the **span** of the vector space defined by v1→...

jelentetvq

jelentetvq

Answered

2022-01-24

Let v1=[23] and v1=[46] what is the **span** of the vector space defined by v1 and v1? Explain your answer in detail?

Answer & Explanation

Gordon Stephens

Gordon Stephens

Expert

2022-01-25Added 10 answers

span ({v1,v2})={λv1λF}
Explanation: Typically we talk about the span of a set of vectors, rather than of an entire vector space. We will proceed, then, in examining the span of {v1,v2} within a given vector space.
The span of a set of vectors in a vector space is the set of all finite linear combinations of those vectors. That is, given a subset S of a vector space over a field F, we have
span(S)={i=1kλksknN,siS,λiF}
(the set of any finite sum with each term being the product of a scalar and an element of S)
For simplicity, we will assume that our given vector space is over some subfield F of C. Then, applying the above definition:
span({v1,v2})={i=1kλiviλiF}
={λ1v1+λ2v2λ1,λ2F}
But note that v2=2v1, and so, for any λ1,λ2F
λ1v1+λ2v2=λ1v1+λ2(2v1)=(λ1+2λ2)v1
Then, as any linear combination of {v1andv2} can be expressed as a scalar multiple of {v1, and any scalar multiple of {v1 can be expressed as a linear combination of {v1andv2} by setting λ2=0 we have
({v1,v2})={λv1λF}

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