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Questions about the properties of vector spaces and linear transformations, including linear systems in general.

23 votes
2 answers
2k views

Formula expressing symmetric polynomials of eigenvalues as sum of determinants

The trace of a matrix is the sum of the eigenvalues and the determinant is the product of the eigenvalues. The fundamental theorem of symmetric polynomials says that we can write any symmetric polynom …
Jules's user avatar
  • 493
1 vote

How many parameters are needed to specify a k-dimensional subspace of R^d?

Here is a more geometric way to find the answer. We can specify a $k$ dimensional subspace by giving $k$ vectors in that subspace. That would require $kn$ parameters, but we are overspecifying. By how …
Jules's user avatar
  • 493
7 votes
1 answer
500 views

Numerical linear algebra: how to compute $b^TA^{-1}b$ efficiently

What is the most efficient way to compute $b^TA^{-1}b$ for a given $A$ and $b$? Do we have to calculate $A^{-1}b$, or is this not necessary? edit: I forgot to mention that A is symmetric and positiv …
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  • 493