The vectors ((3),(2)) and ((-4),(1)) can be written as linear combinations of u and w: ((3),(2))=5u+8w ((-4),(1))=−3u+w. The vector ((5),(-2)) can be written as the linear combination au+bw. Find the ordered pair (a,b).

linnibell17591

linnibell17591

Answered question

2022-11-05

The vectors ( 3 2 ) and ( 4 1 ) can be written as linear combinations of u and w:
( 3 2 ) = 5 u + 8 w ( 4 1 ) = 3 u + w .
The vector ( 5 2 ) can be written as the linear combination au+bw. Find the ordered pair (a,b).
I've tried to eliminate u by multiplying the first equation by 3, the second equation by 5, then adding, but it only leads to w = ( 11 29 11 29 ) . I feel like the algebra from here would be too complicated for what the people who wrote the problem were intending, so perhaps I'm going down the wrong path. Would there instead be a convenient way to manipulate the terms to eventually get ( 5 2 ) on the LHS?

Answer & Explanation

h2a2l1i2morz

h2a2l1i2morz

Beginner2022-11-06Added 19 answers

Here is how I would do it.
( 3 2 ) 2 ( 4 1 ) = ( 11 0 )
Allowing us to find one of the princicipal component vectors of the standard basis in terms of the {u,w} basis.
5 u + 8 w 2 ( 3 u + w ) = ( 11 0 ) 11 u + 6 w = ( 11 0 ) ( 1 0 ) = u + 6 11 w
With that we can say:
( 5 2 ) = 8 ( 1 0 ) ( 3 2 )
And write our vector in terms of this basis.
( 5 2 ) = 8 ( u + 6 11 w ) ( 5 u + 8 w ) = 3 u 40 11 w

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