Study whether the vectors v1(1, 1, 2), v2(2, 3, 0), v3(0, 1, 2) in R 3 are linearly independent.
1
Expert's answer
2022-01-25T17:08:57-0500
A sequence of vectors v1,v2,...,vk from a vector space V is said to be linearly dependent, if there exist scalars a1,a2,...,ak not all zero, such that
a1v1+a2v1+...+akvk=0
where 0 denotes the zero vector.
Consider the set of vectors v1=(1,1,2),v2=(2,3,0),v3=(0,1,2) then the condition for linear dependence seeks a set of non-zero scalars, such that
⎣⎡112230012⎦⎤⎣⎡a1a2a3⎦⎤=⎣⎡000⎦⎤
Augmented matrix
⎣⎡112230012000⎦⎤
R2=R2−R1
⎣⎡102210012000⎦⎤
R3=R3−2R1
⎣⎡10021−4012000⎦⎤
R1=R1−2R2
⎣⎡10001−4−212000⎦⎤
R3=R3+4R2
⎣⎡100010−216000⎦⎤
R3=R3/6
⎣⎡100010−211000⎦⎤
R1=R1+2R3
⎣⎡100010011000⎦⎤
R2=R2−R3
⎣⎡100010001000⎦⎤
Then a1=a2=a3=0.
Therefore the vectors v1=(1,1,2),v2=(2,3,0),v3=(0,1,2) in R3 are linearly independent.
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