Equivalent Statements for Vector Subspace Dimension One Less

Theorem
Let $K$ be a field.

Let $M$ be a subspace of the $n$-dimensional vector space $K^n$.

The following statements are equivalent:


 * $(1): \quad \map \dim M = n - 1$
 * $(2): \quad M$ is the kernel of a nonzero linear form
 * $(3): \quad$ There exists a sequence $\sequence {\alpha_n} $ of scalars, not all of which are zero, such that:
 * $M = \set {\tuple {\lambda_1, \ldots, \lambda_n} \in K^n: \alpha_1 \lambda_1 + \cdots + \alpha_n \lambda_n = 0}$

Also, suppose the above hold.

Let $\sequence {\beta_n}$ be a sequence of scalars such that:
 * $M = \set {\tuple {\lambda_1, \ldots, \lambda_n} \in K^n: \beta_1 \lambda_1 + \cdots + \beta_n \lambda_n = 0}$

Then there is a non-zero scalar $\gamma$ such that:
 * $\forall k \in \closedint 1 n: \beta_k = \gamma \alpha_k$

Proof
Let $M^\circ$ be the annihilator of $M$.

Let $N = M^{\circ}$.

By Results Concerning Annihilator of Vector Subspace, $N$ is one-dimensional and $M = \map {J^{-1} } {N^\circ}$.

Let $\phi \in N: \phi \ne 0$.

Then $N$ is the set of all scalar multiples of $\phi$.

Because:
 * $\map {J^{-1} } {N^\circ} = \set {x \in K^n: \forall \psi \in N: \map \psi x = 0}$

it follows that $\map {J^{-1} } {N^\circ}$ is simply the kernel of $\phi$.

Hence $(1)$ implies $(2)$.

By Rank Plus Nullity Theorem, $(2)$ also implies $(1)$.

Suppose $\sequence {\alpha_n}$ is any sequence of scalars.

Let $\sequence {e'_n}$ be the ordered basis of $\paren {K^n}^*$ dual to the standard ordered basis of $K^n$.

Let $\displaystyle \phi = \sum_{k \mathop = 1}^n \alpha_k e'_k$.

Then, by simple calculation:
 * $\map \ker \phi = \set {\tuple {\lambda_1, \ldots, \lambda_n}: \alpha_1 \lambda_1 + \cdots + \alpha_n \lambda_n = 0}$

It follows that:
 * $\phi \ne 0 \iff \exists k \in \closedint 1 n: \alpha_k \ne 0$

Thus $(2)$ and $(3)$ are equivalent.

Suppose $M = \map \ker \psi$, where $\displaystyle \psi = \sum_{k \mathop = 1}^n \beta_k e'_k$.

Then $\psi = M^\circ$.

As $M^\circ$ is one-dimensional and since $\psi \ne 0$, it follows that:
 * $\exists \gamma \ne 0: \psi = \gamma \phi$

Therefore:
 * $\forall k \in \closedint 1 n: \beta_k = \gamma \alpha_k$