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Tag Archives: linear-algebra
The characteristic polynomial of a product
Let be an infinite field. Then, the affine space is irreducible in the Zariski topology (this follows since the ideal is prime): in other words, it is not the union of two proper algebraic subvarieties. This has the following consequence: … Continue reading
Density of diagonalizable matrices
Consider regarded as a metric space (for instance, identifying it with ). We will prove that the set of diagonalizable matrices is dense in . Let . We want to find a sequence of diagonalizable matrices such that . Suppose … Continue reading
Eigenvalues of the adjoint representation
This is exercise 1.7 from Humphreys’ Introduction to Lie algebras and representation theory. Let be a matrix with distinct eigenvalues . Then the eigenvalues of the adjoint representation are precisely the differences . In fact, suppose is an eigenvector for ; … Continue reading
There exist subsets of R^n in general position of any finite cardinality
Recall that a subset is said to be in general position if any subset of of cardinality less than or equal to is affinely independent. We’ll prove that given any we can find a set of cardinality in general position. … Continue reading
A basis for C[0,1] has the cardinality of the continuum
If is a basis for , then clearly , since . The last inequality follows from the fact that a continuous function with domain is determined by its values over . To prove the other inequality, it suffices to find … Continue reading
Square roots of primes are linearly independent over the rationals
Let be any enumeration of the primes. We’ll prove that by induction on . The cases are straightforward. Let . Suppose now that our claim holds for and , that is, and . We now want to see that ; … Continue reading
A finite dimensional vector space over an uncountable field is not the countable union of proper subspaces
Let be an uncountable field, a finite dimensional -vector space. Without loss of generality ; then every hyperplane is a solution of some linear equation , where is the standard basis for . We now consider the uncountable set . … Continue reading
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