[SOLVED] Linear algebra; conditions for spaces 1. The problem statement, all variables and given/known data 1) If I want to write a basis for R^3, what must the conditions for the three vectors? Must they be linearly independant or orthogonal or what? 2) If I want to write a basis for a supspace of R^3, what must the conditions for the vectors be? Must they only be linearly independant? 3) If I have a symmetric matrix A, I can find an orthogonal matrix S that satisfies S^(-1) *D*S, where D is a diagonal matrix consisting of eigenvalues. If I have a diagonalizable matrix B, which is not symmetric, can I also find an orthogonal matrix S that satisfies what I wrote above? 3. The attempt at a solution For the vectors spanning R^3, I believe they have to be orthogonal and linearly independant. For the vectors spanning a subspace of R^3, I believe the vectors just have to be linearly independant. Correct?