Concept of the Day
Rank–Nullity Theorem
For a linear map T : V → W where V is finite-dimensional, dim V = rank(T) + nullity(T). Here rank(T) = dim of the range of T and nullity(T) = dim of the kernel of T.
Example
T : ℝ³ → ℝ², T(x, y, z) = (x, y). The range is all of ℝ², so rank = 2. The kernel is the z-axis {(0, 0, z)}, so nullity = 1. Check: 3 = 2 + 1.