Why the nontrivial nullspace of a functional has codimension 1?

The nullspace of a linear functional that is not $\equiv 0$ is a linear subspace of codimension $1$.

I don't understand this statement on page 57, Functional Analysis(Pater Lax). Does it mean the dimension of nullspace of a linear functional is either zero or the dimension of the domain of the functional minus one, which I don't see why it's necessarily true.

Added: Thank you all for your valuable comments and answers. I didn't realize that it's wrong to interpret "The nullspace of a linear functional that is not $\equiv 0$" as the nullspace (of a linear functional) that is not $\equiv 0$ until I saw the answers.

Solution 1:

For simplicity, suppose $X$ is a vector space over $\mathbb{R}$, and $f$ a linear functional on $X$.

If $f=0$, then $\ker f = X$, so the codimension is zero.

If $f \neq 0$, then there is some $x_0$ such that $f(x_0) \neq 0$. Now consider the quotient space $Q={X}/{\ker f}$ (ie, two points $x_1,x_2$ are equivalent iff $f(x_1) = f(x_2)$, which basically 'flattens' $X$ 'down' to the values of $f$, ie $\mathbb{R}$).

Pick some $q \in Q$, then we must have $q = \{y\}+\ker f$ for some $y$. Note that by linearity we have $f(y+\frac{-f(y)}{f(x_0)}x_0) = 0$, hence $q = \frac{f(y)}{f(x_0)}(\{x_0\}+ \ker f)$. Hence $\{ \{x_0\}+ \ker f \}$ is a basis for $Q$, hence $\ker f$ has codimension one.

Solution 2:

If your linear functional is $f\colon V \to \mathbb R$ (or whatever field you're working over) it means the nullspace has dimension $\dim V$ if $f = 0$ and $\dim V - 1$ otherwise.

The reason why is the rank nullity theorem: $\dim\operatorname{rank}f + \dim\operatorname{nullity}f = \dim V$. The rank of $f$ is $0$ if $f = 0$. If $f \neq 0$ then the rank is $1$.

Edit: As Yuan points out the above argument only works in the finite dimensional case. For the infinite dimensional case take a basis of the nullspace and extend to a basis of $V$. You can add at most $1$ additional vector because if $f(v) = a$ and $f(w) = b$ then $f(bv - aw) = 0$.