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

Question

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

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Kylie 7 months 2021-10-07T21:05:02+00:00 1 Answer 0 views 0

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    2021-10-07T21:06:04+00:00

    \displaystyle\huge\red{\underline{\underline{Solution}}}

    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.

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