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https://ocw.mit.edu/courses/physics...g-2016/lecture-notes/MIT8_04S16_LecNotes5.pdf

On page 6, it says,

"

Matrix mechanics, was worked out in 1925 by Werner Heisenberg and clarified by Max Born and Pascual Jordan. Note that, if we were to write xˆ and pˆ operators in matrix form,they would require infinite dimensional matrices. One can show that there are no finite size matrices that commute to give a number times the identity matrix, as is required from (2.13). This shouldn’t surprise us: on the real line there are an infinite number of linearly independent wavefunctions, and in view of the correspondences in (2.14) it would suggest an infinite number of basis vectors. The relevant matrices must therefore be infinite dimensional.

"

Equation 2.13: [x, p] = ihbar

2.14 Correspondences:

operators matrices

wavefunctions vectors

eigenstates eigenvectors

The part that I don't get is, "on the real line, there are an infnite number of linear independent wavefunctions." Why is it so? If I think of the real line and think of wavefunctions as vectors, every single vector will be linearly *dependent* not independent, right? What am I missing here?