Linear algebra; conditions for spaces

[Niles]

[SOLVED] Linear algebra; conditions for spaces

Homework Statement

1) If I want to write a basis for R^3, what must the conditions for the three vectors? Must they be linearly independent 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 independent?

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?

The Attempt at a Solution

For the vectors spanning R^3, I believe they have to be orthogonal and linearly independent.

For the vectors spanning a subspace of R^3, I believe the vectors just have to be linearly independent. Correct?

 

[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.

 

[Niles]

Cool, thanks :-)