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Thread: Show that S in linearly Independent and how to extend S to a basis for V

  1. #1
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    Show that S in linearly Independent and how to extend S to a basis for V

    Let
    V={(x$\displaystyle _1$,x$\displaystyle _2$,x$\displaystyle _3$,x$\displaystyle _4$,x$\displaystyle _5$) $\displaystyle \in$R$\displaystyle ^5$: x$\displaystyle _1$-2x$\displaystyle _2$+3x$\displaystyle _3$-x$\displaystyle _4$+2x$\displaystyle _5$=0}.

    (a) Show that S={(0,1,1,1,0)} is a linearly independent subset of V.

    (b) Extend S to a basis for V.
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  2. #2
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    Quote Originally Posted by tn11631 View Post
    Let
    V={(x$\displaystyle _1$,x$\displaystyle _2$,x$\displaystyle _3$,x$\displaystyle _4$,x$\displaystyle _5$) $\displaystyle \in$R$\displaystyle ^5$: x$\displaystyle _1$-2x$\displaystyle _2$+3x$\displaystyle _3$-x$\displaystyle _4$+2x$\displaystyle _5$=0}.

    (a) Show that S={(0,1,1,1,0)} is a linearly independent subset of V.
    A set containing a single non-zero vector is always a linearly independent subset! If av= 0 and v is not 0 then you must have a= 0. Do you see how that verifies the definition of "linearly independent"? Here, all you need to do is show that (0, 1, 1, 1, 0) is in V. $\displaystyle x_1- 2x_2+ 3x_3- x_4+ 2x_5= 0$ becomes 0- 2(1)+ 3(1)- 1+ 2(0)= 0. Is that true?

    (b) Extend S to a basis for V.
    "$\displaystyle x_1- 2x_2+ 3x_3- x_4+ 2x_5= 0$" is a single linear equation in 5 unknown values. You can solve for any one of them, say, $\displaystyle x_4= x_1- 2x_2+ 3x_3+ 2x_5$, leaving you "4 degrees of freedom"- that is your subspace has dimension 4 and a basis will require 4 vectors. Since you are already given one, you need to find 3 more that form an independent set.
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