Gaussian elimination help

Question:The spanning set

[(2,2,1), (4,1,0), (2,-1,-1), (2,0,1)]

of R3 is not independent.

Explain why, and identify a

maximal independent set among these vectors.

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I'm having trouble with this question and i was wondering if someone could help me out and show me what I have to do to answer this question.

Re: Gaussian elimination help

Quote:

Originally Posted by

**kashmoneyrecord** Question:The spanning set

[(2,2,1), (4,1,0), (2,-1,-1), (2,0,1)]

of R3 is not independent.

Explain why, and identify a

maximal independent set among these vectors.

-----------------------------------------------------

I'm having trouble with this question and i was wondering if someone could help me out and show me what I have to do to answer this question.

Well the dimension of $\displaystyle \mathbb{R}^3$ is three, so any set of four vectors is going to have to be linearly dependent.

Let $\displaystyle v_1=(2,2,1),v_2=(4,1,0),v_3=(2,-1,-1),v_4=(2,0,1)$. Then notice that

$\displaystyle e_2=\frac{1}{2}(v_1-v_4)$,

$\displaystyle e_1=\frac{1}{4}(v_1+v_3-e_2)$, and

$\displaystyle e_3=v_4-2e_1$,

where $\displaystyle e_1,e_2,e_3$ are the standard basis vectors. So $\displaystyle \{v_1,v_3,v_4\}$ spans $\displaystyle \mathbb{R}^3$ and hence must be linearly independent.

Re: Gaussian elimination help

yo bro is that suffiencient enough to answer the question or is there more to it? and what does maximal independent set mean?

Re: Gaussian elimination help

okay got it thanks a lot :)