Do the Vectors Span R3? | Linear Algebra Exercises

We determine if a given set of vectors spans R3 by setting up a matrix equation and determining if the coefficient matrix has a nonzero determinant. In one case, we find there is a nonzero determinant and so the system of equations is consistent for any vector in R3, and we have that the given set spans R3. In the other case, we will find R3 must contain vectors which are not a linear combination of the given vectors. #linearalgebra
Spanning Sets: (coming soon)
Linear Algebra course: • Linear Algebra
Linear Algebra exercises: • Linear Algebra Exercises
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[@danagofure1460]

I found your video very helpful, and it explain what I am looking for.

[@WrathofMath]

Glad it was helpful!

[@henrywoo1668]

Thank you for your great video ❤

[@WrathofMath]

My pleasure, thanks for watching!

[@emresar983]

Great video thank you

[@WrathofMath]

Glad it was helpful! Thanks for watching!

[@michael654]

what if the matrix isn't a square so you can't find the determinant? Like if you were asked if S = {(1,2), (3,4), (5,6)} spans R2?

[@WrathofMath]

Then construct the matrix with those vectors as its rows and use Gauss-Jordan elimination. The number of nonzero rows in the RREF is the number of linearly independent rows (and hence the number of linearly independent vectors in the original set). Then, if we end up with 2 nonzero rows -> yes the vectors span R2, if we end up with 1 nonzero row -> no the vectors don't span R2. I go over examples like this in this members only video, but that's the gist: th-cam.com/video/1ydcSGY0utU/w-d-xo.html