Linear algebra; conditions for spaces

1. Jan 6, 2008

Niles

[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?

Last edited: Jan 6, 2008
2. Jan 6, 2008

Dick

The conditions for the vectors in a basis is that they be linearly independent and span the required space. They don't have to be orthogonal unless the the problem says 'orthogonal basis'. Symmetric real matrices have the special property that eigenvectors corresponding to different eigenvalues are orthogonal. That's why you can construct an orthogonal matrix S. For a general diagonalizable matrix B, this is not true.

3. Jan 6, 2008

Niles

Cool, thanks :-)