Let's say I have 3 vectors that are all in R3. If I can use Gaussian elimination to get these vectors into reduced row echelon form, it is to my understanding that this mean that the set spans R3. Does this also mean that the set is a basis for R3?
2026-05-16 01:46:43.1778896003
If I can get a set with n vectors into reduced row echelon form, doesn't that mean that the set is a basis for Rn?
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You are correct. As $R^3$ has dimension 3, any set of three vectors $\{v_1,v_2,v_3\}$ that spans $R^3$ must also be linearly independent. Thus as your three vectors span $R^3$, they must also be linearly independent. Hence they must form a basis for $R^3$.