Let v_1=((1),(1),(0),(0)), v_2=((0),(1),(1),(0)), v_3=((0),(0),(1),(1)), v_4=((2),(0),(0),(1))RR^4 vectors. Show that every v in RR^(4 xx 1) can be written as vectors (v_1,v_2,v_3,v_4) linear combination.

Tiana Hill 2022-09-05 Answered
Let v 1 = ( 1 1 0 0 ) , v 2 = ( 0 1 1 0 ) , v 3 = ( 0 0 1 1 ) , v 4 = ( 2 0 0 1 ) R 4 vectors.
Show that every v R 4 × 1 can be written as vectors ( v 1 , v 2 , v 3 , v 4 ) linear combination.
My attempt:
[ 1 0 0 2 v 1 1 1 0 0 v 2 0 1 1 0 v 3 0 0 1 1 v 4 ]
Where do I go from here? Every input is appreciated.
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Answers (2)

beshrewd6g
Answered 2022-09-06 Author has 12 answers
As det A = 1 0, given any vector B = ( b 1 , b 2 , b 3 , b 4 ), we solve the linear system A X = B with
A = ( 1 0 0 2 1 1 0 0 0 1 1 0 0 0 1 1 ) ; X = ( x 1 , x 2 , x 3 , x 4 )
and thank to Cramer's theorem we get as unique solution the components X of B in the base ( v 1 , v 2 , v 3 , v 4 )
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Robaffonadorkdh
Answered 2022-09-07 Author has 1 answers
| 1 0 0 2 1 1 0 0 0 1 1 0 0 0 1 1 | = | 1 0 0 1 1 0 0 1 1 | | 0 0 2 1 1 0 0 1 1 | = 1 + | 0 2 1 1 | = 1 0
so the four vectors are linearly independent, so they span R 4
so any vector in R 4 can be expressed as a linear combination of them.
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