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I'm not really sure how to extend a basis. I'm trying to do the following question.

Consider the subspace $ W = \{(x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1 = -x_4, x_2 = x_3\}$ of $ \mathbb{R}^4$. Extend the basis $\{(0,2,2,0),(1,0,0,-1)\}$ of $W$ to a basis of $ \mathbb{R}^4$.

I know I need to add another two vectors for it to be a basis of $ \mathbb{R}^4$ but I'm not sure how to pick the vectors. In general, how do you expand a basis?

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3 Answers 3

up vote 2 down vote accepted

Hint: Any $2$ additional vectors will do, as long as the resulting $4$ vectors form a linearly independent set. Many choices! I would go for a couple of very simple vectors, check for linear independence. Or check that you can express the standard basis vectors as linear combinations of your $4$ vectors.

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I seem to be doing a lot of trial and error to find two additional vectors that are linearly independent of the current two. What are we actually doing when we expand a basis? The result is a basis for a new vector space, but do the restrictions in the "current" subspace have any relevance on the new basis? I tried the basis $\{(0,2,2,0), (1,0,0,-1), (1,0,0,0), (0,0,0,1)\}$ but that wasn't linear independent. How would you go about choosing the additional vectors? –  user1520427 Aug 23 '12 at 5:48
    
Try $(1,0,0,0)$ and $(0,1,0,0)$ (plus the two you were given). –  André Nicolas Aug 23 '12 at 5:49
    
Hmm I think I get it now. My one didn't work because the $(1,0,0,-1)$ was a linear combination of $\{(1,0,0,0), (0,0,0,1)\}$ correct? –  user1520427 Aug 23 '12 at 5:51
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@user1520427: Yes, that was the problem. Almost everything works, you had bad luck! –  André Nicolas Aug 23 '12 at 5:55

You might take a different 2-D subspace $X = \{(x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1 = x_4, x_2 = -x_3\}$ which has a trivial intersection with $W$ and find a basis for it, for example $\{(0,2,-2,0),(1,0,0,1)\}$.

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and you can always find an example of such a space as the orthogonal complement, here the kernel of $\begin{pmatrix} 0 & 2 & 2 & 0 \\ 1 & 0 & 0 & -1 \end{pmatrix}$ –  Cocopuffs Aug 23 '12 at 5:23

Write the given vectors with $e_1, e_2, e_3, e_4$, the standard basis of $\mathbb{R}^4$ in the columns of a matrix. By some row operations you will have four linearly independent vectors that will be basis.

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For some basic information about writing math at this site see e.g. here, here, here and here. –  Julian Kuelshammer Dec 18 '12 at 23:05
    
Could you specify what row operations you mean? I don't think the answer given is very clear (it may just be me!) –  Tom Oldfield Dec 18 '12 at 23:09

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