Find the value(s) of h for which the vectors are linearly dependent. Justify eac

Tyra 2021-11-08 Answered
Find the value(s) of h for which the vectors are linearly dependent. Justify each answer.
[153],[296],[3h9]
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Expert Answer

un4t5o4v
Answered 2021-11-09 Author has 105 answers
Step 1
Definition: A set of vectors  {v1,..,vn} is linearly dependent
x1=x2==xn=0  is NOT the only solution to the equation
x1v1+x2v2++xnvn=0     (1)
Of course  x1=x2==xn=0  will always be one solution to this equation, but it may or may not be the only solution.
Step 2
Notice that in this case, equation (1) is equivalent to the equation  Ax=0
A=[12359h369]
x=[x1x2x3]
Step 3
We can find the solution to  Ax=0  by row reduction:
[123|059h|0369|0][123|001h15|0000|0]  (R2R25R1R3R1+2R2)
[1027+2h|001h15|0000|0]      (R1R1+2R2)
Step 4
From here we see that for any value of h, the possible solutions of Ax=0 are
[x1x2x3]=[(272h)x3(15h)x3x3]=x3[272h15h1]
Since the last coordinate can be anything and is independent of h , we see that regardless of the value of h, this set of vectors is always linearly dependent.
Result:
The vectors are linearly dependent for all h
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