Subspaces
Chapter 3: Subspaces and Orthogonality — Orthogonality & Rank-Nullity
From the book
Chapter 3: Subspaces and Orthogonality. In the chapter mind map this icon labels Orthogonality & Rank-Nullity. The discussion below is excerpted and lightly edited from § Theorem: Rank-Nullity Theorem in Mathematics for AI and Machine Learning.
For any matrix $A \in \mathbb{R}^{M \times N}$ with rank $r$:
The Rank-Nullity Theorem states that the dimension of the column space (the "range" of the transformation) plus the dimension of the null space (the "kernel") equals the number of columns. This theorem has profound implications: if a matrix has many independent columns (high rank), it has a small null space: few input directions are lost, and $A$ keeps most inputs distinguishable. Conversely, if the rank is low, the null space is large, meaning many inputs map to zero.
What this drawing shows
What you see. Shows nested or intersecting linear spaces, representing spans closed under addition and scalar multiplication.
In the mind map. Chapter 3 — Orthogonality & Rank-Nullity. See From the book above for definitions, figures, and worked examples.
Where to read next
Read the full definitions, figures, and worked examples in Chapter 3: Subspaces and Orthogonality — see the mind-map node Orthogonality & Rank-Nullity.
