The dimension n is, loosely speaking, the number of different things you could observe after making a measurement on the particle.
Let V be an 𝔽-vector space (where 𝔽 is either ℚ or ℝ or ℂ) and let n be a positive interger. If there is a list { v_{1}, v_{2}, … , v_{n} } of vectors that is a basis for V, then V is n-dimensional (or, Vhas dimension n). The zero vector space has dimension zero. If V has dimension n for some nonnegative integer n, then V is finite dimensional; otherwise, V is inifinite dimensional. If V is finite dimensional, its dimension is denoted by dim V.
Theorem 1:
Let V be an n-dimensional vector space, and let { v_{1}, v_{2}, … , v_{n} } be any bssis.
If a set in V has more than n vectors, then it is linearly dependent.
If a set in V has fewer than n vectors, then it does not span V.
Let n = dim V and k = dim ker(T), so 0 ≤ k ≤
n. If n = 0 or if k = n, there is nothing to prove, so we
may assume that 0 ≤ k < n.
If k = 0, let α = {w_{1},
w_{2}, ... , w_{n}} be a basis for V.
Then
then c_{1}w_{1} + c_{2}w_{2} + ... + c_{n}w_{n} ∈ ker(T)
= {0}, hence, the linear independence of α implies that
c_{1} = c_{2} = ... = c_{n} = 0
and so β is linearly independent.
If k ≥ 1, let {v_{1},
v_{2}, ... , v_{k}} be a basis for ker(T), which we extend to a basis
so the linear independence of α implies that c_{1} =
c_{2} = c_{n-k} = a_{1} =
a_{2} = ... = a_{k} = 0. We conclude that β is linearly independent.
Corollary:
Let V and U be finite dimensional vector spaces over the same
field of scalars (either real numbers or complex numbers). Suppose that
dimV = dim U and let T be a linear transformation from
V into U. Then ker(T) = {0} if and only if the
range of T is U.
If ker(T) = {0}, then dim ker(T) = 0 and
dim range(T) = dimV = dim U. But the image of T is
a subspace of U; hence range(T) = U.
Conversely, if range(T) = U, then dim range(T) =
dimV = dim U and we get dim ker (T) = 0, so ker(T)
= {0}.
Corollary (The dimension Theorem for matrices):
Let A be an m×n matrix. Then
\[
\mbox{dim}\,\mbox{nullspace}({\bf A}) + \mbox{dim}\,\mbox{column}({\bf A}) =
n \quad (\mbox{the number of columns in }\,{\bf A} .
\]
If m = n, then the nullspace of A is {0} if and
only if it is a full range matrix.
Apply the preceding theorem to the linear transformation generated by matrix
A.