- #1

fog37

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Aside from a rigorous definitions, a linear vector space contains an infinity of elements called

**vectors**that must obey certain rules. Based on the dimension ##N## (finite or infinite) of the vector space, we can always find a set of ##n=N## linearly independent vectors that can form a basis. For each vector space, there is an infinity of possible bases to choose from. The basis vectors inside a particular basis don't need to be orthogonal or unit in length. Among the many many bases there is one specific basis that is orthonormal: it is composed of unit vectors that are pairwise orthogonal to each other. Among all the possible bases, only one is orthogonal and orthonormal, correct? Or are there multiple orthonormal bases?

A subspace of a vector space is also a set that contains an infinite number of vectors but it contains "less" vectors than the original host vector space. What can we say about the bases of a subspace ##B## of a vector space ##A##? For example, if the host vector space ##A## has ##N=4##, it means that:

- Each vector in ##A## has four components: ##a = (a_{1}, a_{2}, a_{3}, a_{4})##
- Each possible basis contains 4 linearly independent vectors

Subspace ##B##, being a vector space on its own, also has an infinity of possible bases but only one specific orthonormal basis, right?

Thanks!