# ML Wiki

## Four Fundamental Subspaces

A matrix $A$ has four subspaces:

• Column Space $C(A)$ or $\text{ran}(A)$: Range of $A$
• Nullspace $N(A)$ or $\text{null}(A)$
• Row Space $C(A^T)$ of $A$ is the same as Column Space of $A^T$
• Nullspace of $A^T$ (also called "Left Nullspace")

## Some Properties

Suppose we have an $m \times n$ matrix of rank $r$

### Orthogonality

• Nullspace of $A$ is orthogonal to the row space: $N(A) \; \bot \; C(A^T)$
• Left nullspace of $A$ is orthogonal to the column space: $N(A^T) \; \bot \; C(A)$
• see the proof in Space Orthogonality#Row space and Nullspace

## The Four Spaces

### Column Space

• $\text{dim } C(A) = r$, there are $r$ pivot columns
• basis: columns of $A$
• also called "range"
• $\text{ran}(A) = \{ \mathbf y \in \mathbb R^m \ : \ \mathbf y = A \mathbf x \ \forall \mathbf x \in \mathbb R^n \}$
• if we think about Linear Transformation $T_{A}$ formed by $A$, this is what it does to an $n$-dimensional vector $\mathbf x$: it brings it to an $m$-dinesional vector $\mathbf y$
• all such vectors $\mathbf y$ form the column space of $A$

### Row Space

• $\text{dim } C(A^T) = r = \text{dim } C(A)$, there are $r$ pivot rows - the same dim as for Column Space
• Let $R$ be Row Reduced Echelon Form of $A$, then $C(A^T) = C(R^T)$
• basis: first $r$ rows of $R$

### Nullspace

• $\text{dim } N(A) = n - r$ - the number of free variables
• basis: special solutions for $A\mathbf x = \mathbf 0$
• $N(A) = \text{null}(A) = \{ \mathbf x \in \mathbb R^n \ : \ A \mathbf x = \mathbf 0 \}$

### Left Nullspace

• This is the nullspace of $A^T$ ($A^T$ is $n \times m$ matrix of rank $r$)
• $\text{dim } N(A^T) = m - r$ - there are $m$ columns, $m$ variables, and $m - r$ free variables

## Singular Value Decomposition

We know how to find the basis for all the subspaces

SVD finds these bases:

• if $A V = U \Sigma$ then
• $\mathbf v_1, \ ... \ , \mathbf v_r$ is the basis for the row space $C(A^T)$
• $\mathbf v_{r+1}, \ ... \ , \mathbf v_{n}$ is the basis for the nullspace $N(A)$
• $\mathbf u_1, \ ... \ , \mathbf u_r$ is the basis for the column space $C(A)$
• $\mathbf u_{r+1}, \ ... \ , \mathbf u_{m}$ is the basis for the left nullspace $N(A^T)$