The vector x is in H = Span {v_{1}, v_{2}} and find the beta-coordinate vector [x]_{beta}

usagirl007A 2021-01-25 Answered
The vector x is in H=Span v1,v2
and find the beta-coordinate vector [x]β
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Expert Answer

likvau
Answered 2021-01-26 Author has 75 answers

Let vector x is in a vector space V and β=b1,b2,,bn is a basis for V,
then the beta- coordinates of x are the weights c1,c2,...,cn or [x]β=β[c1c2cn] such that x=c1b1+c2b2++cnbn.
Given:
The vectors v1=[115107],v2[1481310],and x=[19131815]
Calculation:
Here β=v1,v2 and H=Spanv1,v2.
Therefore, for showing x is in H, show that the vector equation x=x1v1+x2v2 has a solution.
Write the given vectors as an augmented matrix [v1v2x] and find the row reduced form of the matrix.
[111419581310131871015](R1R1+R2)R3R3+2R2[666581303871015]
[666581303871015]R116R1[111581303871015][111581303871015](R2R25R1)R4R47R1[111038038038][111038038038](R3R3+R2)R4R4+R2[111038000000]
On further simplification the matrix becomes,

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