69. Rank–Nullity Theorem

For a linear transformation $𝑇:𝑉\to 𝑊$ between finite-dimensional vector spaces:

In matrix form, for an $𝑚\times 𝑛$ matrix $𝐴$:

(See Dimension, Null Space, and Rank.)

The dimension of the input space splits exactly into the part that gets sent to zero (the kernel) and the part that survives as the image.

69.1. Why it’s true (sketch)

Pick a basis $\left\{{𝑘}_1,\dots ,{𝑘}_{𝑝}\right\}$ for $ker\left(𝑇\right)$, then extend it to a basis $\left\{{𝑘}_1,\dots ,{𝑘}_{𝑝},{𝑣}_1,\dots ,{𝑣}_{𝑞}\right\}$ for $𝑉$ (so $𝑛=𝑝+𝑞$).

Then $\left\{𝑇\left({𝑣}_1\right),\dots ,𝑇\left({𝑣}_{𝑞}\right)\right\}$ is a basis for $im\left(𝑇\right)$ — they span (because the $𝑘$’s map to $0$) and are linearly independent (a non-trivial dependency would land back in $ker\left(𝑇\right)$, contradicting linear independence of the $𝑣$’s with the $𝑘$’s).

So $dim\left(im\left(𝑇\right)\right)=𝑞$, and $dim\left(ker\left(𝑇\right)\right)+dim\left(im\left(𝑇\right)\right)=𝑝+𝑞=𝑛$. ∎

69.2. Why it matters

It’s the bookkeeping identity for linear maps. Lots of immediate consequences:

• Injectivity from dimensions: if $𝑇:{ℝ}^{𝑛}\to {ℝ}^{𝑚}$ has $𝑛>𝑚$, then $dim\left(ker\left(𝑇\right)\right)\ge 𝑛-𝑚>0$ — not injective.
• Surjectivity from dimensions: if $𝑇:{ℝ}^{𝑛}\to {ℝ}^{𝑚}$ with $𝑛<𝑚$, then $dim\left(im\left(𝑇\right)\right)\le 𝑛<𝑚$ — not surjective.
• Square invertibility: a square matrix $𝐴$ is invertible iff $ker\left(𝐴\right)=\left\{\mathrm{𝟎}\right\}$ iff $im\left(𝐴\right)={ℝ}^{𝑛}$ — by rank–nullity these are equivalent for square $𝐴$.
• Solution space dimension: the solution set of $𝐴\overrightarrow{𝑥}=\overrightarrow{𝑏}$ (when consistent) has dimension $𝑛-\text{rank}\left(𝐴\right)$ — see Linear System Solutions.

69.3. Equivalent statements

For an $𝑚\times 𝑛$ matrix $𝐴$:

• $\text{rank}\left(𝐴\right)+\text{nullity}\left(𝐴\right)=𝑛$ (number of columns)
• $\text{rank}\left(𝐴\right)=\text{rank}\left({𝐴}^{𝑇}\right)$ — row rank = column rank
• $\text{nullity}\left({𝐴}^{𝑇}\right)=𝑚-\text{rank}\left(𝐴\right)$ — gives the dimension of the left null space

69.4. See also

• Rank
• Null Space / Kernel
• Image / Column Space
• Surjective / Injective / Bijective