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Niles
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[SOLVED] Linear algebra; conditions for spaces
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?
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?
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 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?
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?
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