Two different ways to express begin{bmatrix}11 end{bmatrix} as a linear combination of v_{1}, c_{2} and v_{3}.

Tazmin Horton 2020-11-02 Answered
Two different ways to express [11]
as a linear combination of v1,c2 and v3.
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Expert Answer

Daphne Broadhurst
Answered 2020-11-03 Author has 109 answers
Given vectors are v1=[13],v2=[28] and v3=[37]
Calculation:
Let the vector w = [11].
The vector w can be represented in terms of as w=av1+vv2+cv3.
Determine the coefficients a, b, c.
Write the vectors v1,v2 and v3
as the columns of a matrix [v1v2v3 2] and reduce the matrix in to row reduce echelon form.
[12313871]R23R1+R2[12310224]
[12310224]R212R2[12310112]
[12310112]R1R12R2[10550112]
This gives the following equations,
a5c=5
b+c=2
Here, c is a free variable.
Two find two different ways to write w as a linear combination of v1,v2,v3, choose two different values of c, which will give two different values of corresponding a and b.
Let, c=1, this gives,
a=10
b=3
Then, the vector w can be expressed as w=10v13v2+v3.
Let c=2, this gives,
a=15
b=4
Then, the vector w can be expressed as w=15v14v2+2v3.
Therefore, two different ways of expressing [11]
as a linear combination of v1,v2,v3 are 10v13v2+v3 and 15v14v2+2v3.
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