Basis
The bases for a linear space describe the space. Each member basis is independent.
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Bases
For any linear space, the bases are independent vectors that can be linearly combined to reach every other vector in the space. If a basis is removed, the space necessarily shrinks.
A null space has no basis, but all other spaces have infinitely many possible bases, because the only requirement on a basis is that it be independent.
A convenient pair of basis vectors for R2 space are [1 0] and [0 1].
Change of Basis
Any two independent vectors can form the basis for an R2 space, but [1 0] and [0 1] are the most convenient bases. The linear space can be transformed between bases, to normalize values into a convenient shape.
This linear transformation can be expressed with a matrix; the inverse transformation can also be expressed with the inverse of that same matrix.
The primary example of this is diagonalization, where a matrix is linearly transformed to be a diagonal matrix of eigenvalues.