Questions tagged [hamel-basis]

A Hamel basis (often simply called basis) of a vector space $V$ over a field $F$ is a linearly independent subset of $V$ that spans it.

68 questions
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Why in a finite dimensional space every orthonormal basis is basis

Why in a finite dimensional space every orthonormal basis is basis i know in infinite dimensional space every basis is orthonormal basis but converse is not true ( for example $l^2$ ) but in finite ...
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Open set minus closed set with empty interior

Let $X$ be a separable infinite-dimensional Banach space, $U$ be non-empty and open in $X$ and $E$ be finite-dimensional linear subspace of $X$. I would like to know if there is a non-empty open ...
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Eigendecomposition of rank-deficient rank-$1$ update

I have a matrix of the form $C = I - a a^T$. In this particular case, $a$ is a constant vector with value $1/\sqrt{N}$, where $N$ is the rank of $I$ (number of columns). For $I = eye(5)$, $N$ will be ...
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Does the statement “Every short exact sequence of vector spaces splits” imply the axiom of choice? [duplicate]

Using the axiom of choice, or more directly, the statement that every linearly independent set of vectors in a vector space may be extended to a basis, it is easy to prove that every short exact ...
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Is it true? $\dim(U+W)=\dim(U\cap W)+1$ implies $U\subseteq W$

I'm curious whether or not this statement is true. If $U,W$ are finite-dimensional space satisfying $\dim(U+W)=\dim(U\cap W)+1,$ then $U\subseteq W.$ Any help would be appreciated! Thank you