How to extend a set to form a basis?

[mitch_1211]

i want to extend the set S={(1,1,0,0),(1,0,1,0)} to be a basis for R4. I know I am going to need 4 vectors, so i need to find 2 more that aren't linear combinations of the first 2.

Is there a better way to approach this other than choose 2 at random and check linear independence/dependence until i find 2 that work?

thanks!

 

[DivisionByZro]

Guessing is actually a decent way of doing this problem. It's late, but it seems to me that your two other vectors could be {0,0,1,1},{0,1,0,1}. Try to see if this is a basis for R4. Off the top of my head this seems about right.

 

[Bacle]

In your particular case, a vectort with a nonzero fourth coordinate would work, but
in general, let me think.

 

[td21]

There is no particular method.
Yes, choosing wisely for some vectors that look like to be independent, and then computing the determinant or the rank to check if they are really so.

 

[spamiam]

In general to extend a set to a basis, you find the span of your original set. Pick any vector not in the span (this guarantees linear independence) and use this as a new basis vector, adding it as a member to your original set. Take the span of this new set and use the same process. Rinse and repeat until you've found a basis. (This process will always terminate for finite-dimensional vector spaces.)

In other words, given a vector space $V$ and a set $S = \{v_1, \ldots, v_n\}$, if S doesn't span V, then $V \backslash \text{span}(S)$ is nonempty, so we may choose a linearly independent vector $w_1$ from this set. Form $S_1 = \{v_1, \ldots, v_n, w_1\}$ and once again if $S_1$ doesn't span V then $V \backslash \text{span}(S_1)$ is nonempty, so we may pick another linearly independent vector $w_2$ from this set. We continue this process until we find a basis.

 

[mitch_1211]

thanks everyone, i will just have to evaluate on a case-by-case basis if it is easy enough to guess a few vectors and check or if i need to find the span first like spamiam suggested.

 

[Bacle]

td21:

Do you know for certain that there is no method? Only way I can think of proving
such claim is some argument/proof of impossibility. Also, do you mean there are no
ways of constructing n+1 LI vectors given n LI vectors?

Method-wise, you can always use the fact that a matrix with n LI vectors will
always be similar to the nxn-identity matrix, which contains the standard basis
as columns/rows.