Scanned course notes:

Notes on linear algebra follows {Horn & Johnson, 1990. Matrix Analysis}.

### Four fundamental subspaces

For a linear transformation $A \in \mathbb{R}^{m\times n}$ with rank $r$ and singular value decomposition $A = U \Sigma V^T$, its four fundamental subspaces are summarized in the following table.

Table: Four fundamental subspaces of a linear transformation

Name Notation Alternative name Alternative notation Containing Space Dimension Basis
Image $\mathrm{im}(A)$ Column space $\mathrm{range}(A)$ $\mathbb{R}^m$ $r$ First $r$ columns of $U$
Kernel $\mathrm{ker}(A)$ Null space $\mathrm{null}(A)$ $\mathbb{R}^n$ $n-r$ Last $n-r$ columns of $V$
Coimage $\mathrm{im}(A^\mathrm{T})$ Row space $\mathrm{range}(A^\mathrm{T})$ $\mathbb{R}^n$ $r$ First $r$ columns of $V$
Cokernel $\mathrm{ker}(A^\mathrm{T})$ Left null space $\mathrm{null}(A^\mathrm{T})$ $\mathbb{R}^m$ $m-r$ Last $m-r$ columns of $U$

The fundamental theorem of linear algebra:

1. The kernel is the orthogonal complement of the coimage: $\mathrm{ker}(A)^\perp = \mathrm{im}(A^\mathrm{T})$
2. The cokernel is the orthogonal complement of the image: $\mathrm{ker}(A^\mathrm{T}) = \mathrm{im}(A)^\perp$