4 Fundamental Subspaces - Description (𝐴 is an 𝑚𝗑𝑛 matrix of real-numbers ℝ)

Linear Subspace Types	Syntax	Description
Row Space Image of 𝐴^𝑇 Range of 𝐴^𝑇	𝐶(𝐴^𝑇)	contains all linear combinations of the ROW vectors of 𝐴 (i.e. columns of 𝐴^𝑇) dimension = 𝑟𝑎𝑛𝑘 a subspace of ℝ𝑛 orthogonal to null-space
Null Space Kernel Space	𝑁(𝐴)	contains all solutions to the system 𝐴𝑥̅=𝑜̅ dimension = 𝑛 - 𝑟𝑎𝑛𝑘 (nullity is the dimension of null space) a subspace of ℝ𝑛 orthogonal to row-space
Column Space Image of 𝐴 Range of 𝐴	𝐶(𝐴)	contains all linear combinations of the COLUMN vectors of 𝐴 dimension = 𝑟𝑎𝑛𝑘 a subspace of ℝ𝑚 orthogonal to left-null-space
Left Null Space	𝑁(𝐴^𝑇)	contains all solutions to the system 𝐴^𝑇𝑦̅=𝑜̅ dimension = 𝑚 - 𝑟𝑎𝑛𝑘 a subspace of ℝ𝑚 orthogonal to left column-space

4 Fundamental Subspaces - Diagram (𝐴 is an 𝑚𝗑𝑛 Matrix)

𝐴 transforms vector 𝑥 from ℝ^𝑛 to ℝ^𝑚
𝐴^𝑇 transforms vector 𝑥 from ℝ^𝑚to ℝ^𝑛
row-space and null-space are complements in ℝ^𝑛 space
column-space and left-null-space are complements in ℝ^𝑚space
The row-space is orthogonal to the null-space because 𝐴𝑥 = 0 means the dot product of 𝑥 with each row of 𝐴 is 0
The column-space is orthogonal to the left-null-space because 𝐴^𝑇𝑥 = 0 means the dot product of 𝑥 with each column of 𝐴 is 0