Find a basis in R3 for the subspace (Linear Algebra)?

Simple 3-t

The idea of a basis is to find a set of equations that you can add together to create all the other equations

In your example you have three equations x,y,z. If you look you will notice that y=0-x, z=x+x, so for this problem the answer is x. Actually you can multiply x by any nonzero constant and you will still get the right answer, so y,z are also valid answers.

Equivalently, find the solution space of the homogeneous equation Ax = 0, where A = || 1 2 3 || ......|| 0 1 1 || Subtract 2 times the second row from the first row to put A in row echelon form giving A' = || 1 0 1 || .......|| 0 1 1 || You can now read the general solution as vectors of the form [ -t -t t ]' = t*[ -1 -1 1]'. This exhibits [ -1 -1 1]' as your basis.