1. The problem statement, all variables and given/known data This is the question on my assignment: In each case below, given a vector space V , find a basis B for V containing the linearly independent set S ⊂ B. It has a bunch of different cases but I think that if you help me with the following two, I will learn enough to do the others. The first case is the following: (a) V = R[itex]^{4}[/itex], S = {(1,0,0,1),(0,1,1,0),(2,1,1,1)}. and the next case is: (b) V = M2×2 = the vector space of all 2 × 2 matrices, and S= [1, 1; 1, 0] [0, 1; 1, 1] [1, 0; 1, 1] 2. Relevant equations 3. The attempt at a solution My problem with BOTH cases is this: I only know how to find a basis given a bunch of vectors that form a span (in other words, I know how to find the linearly dependent ones and kick them out of the equation). But I DONT understand how to find the missing parts of the basis given what the basis is SUPPOSED to span. Can someone please walk me through this? And my SECOND problem is with case (b): I cannot visualize how matrices can span something. I understand vectors, but not matrices. And since I can't understand it, I can't approach it to find the basis. Thanks SO much in advance for your help!
For (a), what you can do is solve the system S'*x=0 for a nonzero solution. In other words, you want to find a vector that are orthogonal to the span of columns of S, i.e., in the nullspace of S'. You may start with a random vector, project onto span{S} and subtract the projection, if you're lucky the remainder is nonzero, either the remainder or the original random vector is linearly independent. For (b), all you have to do is "straighten" the 2x2 matrix into a 4x1 vector. As linear spaces they are isomorphic.
OK, I am a bit confused. So solving S'*x-0, S' is the matrix whose columns are the vectors of S right? So when I do that I will be finding the nullspace of that matrix.. When I do this, I get x1, x2, and x3 all equal to 0. Is this right? I'm sorry for my thick headedness.. but linear algebra is by far my weakest class.
No worries, I should've been more explicit. S' is the transpose of S, where S is the 4x3 matrix whose columns are those vectors, so S' is 3x4. Your solution should be a 4x1 vector (x1,x2,x3,x4)≠0, it is orthogonal and therefore independent to rows of S', i.e., columns of S, i.e., your original vectors.