Consider the vectors v=((1),(3)) and w=((3),(2)). If the vectors v and w are linearly independent, answer with 0. If they aren't, find coefficients a and b, not both 0, such that a((1),(3))+b((3),(2))=((0),(0))and answer with ab.

Lisa Hardin

Lisa Hardin

Answered question

2022-07-16

Consider the vectors v = ( 1 3 ) and w = ( 3 2 )
If the vectors v and w are linearly independent, answer with 0. If they aren't, find coefficients a and b, not both 0, such that a ( 1 3 ) + b ( 3 2 ) = ( 0 0 ) and answer with a b
I know that vectors v and w are not linearly independent but im not sure how to get a b

Answer & Explanation

ab8s1k28q

ab8s1k28q

Beginner2022-07-17Added 17 answers

a ( 1 3 ) + b ( 3 2 ) = ( 0 0 ) . This becomes the system of equations
a + 3 b = 0 , 3 a + 2 b = 0.
The first equation gives a=−3b, and substituting that into the second equation gives 3 ( 3 b ) + 2 b = 0 , which becomes 7 b = 0 and therefore, b=0. Substituting back into the expression, we see that a=0.
Damien Horton

Damien Horton

Beginner2022-07-18Added 5 answers

A sequence of vectors { v , w } are linearly independent if the equation
a v + b w = 0
is only satisfied by a=b=0. Therefore, we represent the vectors by the homogenuous linear system
[ 1 3 3 2 ] [ a b ] = [ 0 0 ]
solving for a and b we find
{ a + 3 b = 0 3 a + 2 b = 0 { a = 3 b a = 2 3 b { a = 0 b = 0
therefore the sequence of vectors { v , w } are linearly independent.

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