# Finding a basis given part of that basis

by skyturnred
Tags: basis
 P: 116 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$^{4}$, 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!