Math, asked by Nikhil833441, 9 months ago

Determine whether or not the vectors u(1, 1, 2),ν (2,3, 1),
w(4,5,5) in R3 are linearly dependent.

Answers

Answered by pulakmath007
21

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FORMULA TO BE IMPLEMENTED

LINEARLY DEPENDENT

A set of vectors { u, v, w } are said to be linearly dependent if there exists three scalers a, b, c , not all zero such that

 \sf{ \: au + bv + cw = 0 \: }

CALCULATION

If possible there exists three scalers a, b, c such that

 \sf{ \: au + bv + cw = 0 \: }

 \implies \:  \sf{ a(1, 1, 2)+b(2,3, 1)+c(4,5,5) = ( 0, 0, 0) \: }

 \implies \sf{ (a+2b+4c, a+3b+5c, 2a+b+5c)=(0,0,0)\: }

So

  \sf{ a+2b+4c = 0}  \:  \: ....(1)

 \sf{  a+3b+5c = 0\: } \:  \: ...(2)

  \sf{  2a+b+5c = 0\: } \:  \:  \: .....(3)

Equation (1) - Equation (2) gives

 \sf{ \:  b =  - c\: }

From Equation (1)

 \sf{ \:  a  - 2c + 4c = 0 \: }

 \sf {\implies \: a=  - 2c}

 \sf{ \: Let  \: us  \: suppose \: c = 1 \: }

So

 \sf{ \:  a =  - 2\: }

 \sf{ \:  b =  - 1\: }

Hence from above

 \sf{ \:  - 2u  - v + w = 0 \: }

Thus there exists a non zero set of values of a, b, c such that

 \sf{ \: au + bv + cw = 0 \: }

Hence the vectors u(1, 1, 2),v (2,3, 1),w(4,5,5) are linearly dependent.

Answered by avinashvind986
0

Answer:

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