What disappears, and what can you reach?

Source scope: §3.2.2. Worked numerical examples, experiments, and practice are teaching supplements unless identified as source results.

Collapse a direction

Take $A=\begin{bmatrix}1&1\\0&0\end{bmatrix}$. Then $A(x,y)=(x+y,0)$. Every output lies on the horizontal axis. Inputs $(t,-t)$ disappear entirely.

For a linear map $T:V\to W$, define

The source leaves checking that both are subspaces as an exercise. Teaching verification: $T0=0$; sums and multiples of zero-output inputs still map to zero. For the range, $Tx+Ty=T(x+y)$ and $\alpha Tx=T(\alpha x)$ show closure. Notice that the nullspace lives in the input space; the range lives in the output space.

Source footnote 3: Algebraists also call the nullspace the kernel. The notes avoid that name because machine learning uses “kernel” in another sense.

Read the spaces from a matrix

For $A\in\mathbb R^{m\times n}$, its column space is the span of its columns in $\mathbb R^m$. Since $Ax$ is the columns weighted by coordinates of $x$, this equals the range of its induced map. The source uses $\operatorname{range}(A)$ for that space. Its row space is the span of its rows in $\mathbb R^n$, written $\operatorname{range}(A^T)$.

The row and column spaces have the same dimension. This common value is the rank:

The source states this equality here without proof. For our example, the column space is $\operatorname{span}\{(1,0)\}$, the row space is $\operatorname{span}\{(1,1)\}$, and the rank is one. Equal dimensions do not mean equal spaces.

Explore in three dimensions

Transfer

Can a $2\times3$ matrix have a range equal to $\mathbb R^3$?