01.01.18 · foundations / linear-algebra

Linear manifolds, hyperplanes, and affine subspaces

shipped3 tiersLean: none

Anchor (Master): Shilov *Linear Algebra* Ch. 2; Berger *Geometry I* Ch. 2–3 (affine spaces, affine maps, the affine group, the projective completion); Gallier *Geometric Methods and Applications* Ch. 2–3 (affine and projective geometry); Bourbaki *Algèbre* Ch. II §9 (affine and projective spaces); Artin *Geometric Algebra* Ch. II

Intuition Beginner

A subspace is a flat sheet that must pass through the origin: a line through the centre, a plane through the centre, the whole space. But most of the lines and planes you actually care about do not pass through the origin. The line a thrown ball traces, the plane of a tabletop sitting above the floor — these are flat, but they are shifted away from the centre. A linear manifold is exactly such a shifted flat: take a subspace through the origin and slide the whole thing by one fixed offset.

So every linear manifold is built from two pieces. One piece is a direction: which way the flat runs, a subspace that records the slopes but forgets the position. The other piece is a single anchor point that says where the flat actually sits. Two different anchor points on the same flat give the same direction, because sliding from one point of the flat to another never leaves the flat. The direction is forced by the flat; the anchor is your free choice.

The most important case is a flat that is one dimension short of filling the whole space — a line inside a plane, or a plane inside ordinary space. These are hyperplanes, and they are exactly the things cut out by a single linear equation. The solution set of one equation in three unknowns is a plane; the solution set of one equation in two unknowns is a line. A whole system of equations carves out the flat where all those single-equation flats overlap.

Visual Beginner

Picture a flat plane through the origin, tilted in space, and then picture a second plane parallel to it, floating a fixed distance above. The lower plane is a subspace — it contains the origin. The upper plane is a linear manifold — the same set of directions, but lifted off the origin by one offset arrow. Every point of the upper plane is the tip of that offset arrow plus some arrow lying in the lower plane.

Two things to read off. The two planes share their directions but not their position, so they are parallel. And the upper plane is the set of points where one fixed measurement — the height read along the perpendicular arrow — equals one fixed number. That single measurement equalling a constant is the linear equation that cuts out the flat.

Worked example Beginner

Take ordinary three-dimensional space with coordinates $(x, y, z)$ . Consider the flat $L$ described by the single equation $$ x + y + z = 6. $$ We will write the same flat in two ways: as an anchor point plus directions, and as a level set of one measurement.

Step 1. Find one point on $L$ . Try $x = 6$ , $y = 0$ , $z = 0$ : then $6 + 0 + 0 = 6$ , so the point $(6, 0, 0)$ lies on $L$ . Call it the anchor.

Step 2. Find the directions. A direction is a step you can take from one point of $L$ to another without leaving $L$ . Stepping from $(6, 0, 0)$ to $(5, 1, 0)$ keeps the sum at $6$ , so $(- 1, 1, 0)$ is a direction. Stepping to $(5, 0, 1)$ also keeps the sum at $6$ , so $(- 1, 0, 1)$ is a second direction. These two arrows are not multiples of each other, so they span a plane of directions.

Step 3. Write the flat both ways. As anchor plus directions, $$ L = (6, 0, 0) + s,(-1, 1, 0) + t,(-1, 0, 1), $$ where $s$ and $t$ range over all numbers. As a level set, $L$ is the set of points where the measurement "add the three coordinates" returns the number $6$ .

Step 4. Check. Put $s = 1$ , $t = 2$ : the point is $(6, 0, 0) + (- 1, 1, 0) + (- 2, 0, 2) = (3, 1, 2)$ , and $3 + 1 + 2 = 6$ . The point lands on $L$ , as it must.

What this tells us: one linear equation in three unknowns describes a plane, and that plane is an anchor point plus a two-dimensional sheet of directions. The equation form and the anchor-plus-directions form are two descriptions of one flat.

Check your understanding Beginner

Formal definition Intermediate+

Let $K$ be a field and $V$ a $K$ -vector space, with subspaces, bases, and dimension as in 01.01.04 and linear functionals and the dual space $V^{*}$ as in 01.01.02.

Definition (linear manifold / affine subspace). A linear manifold (equivalently affine subspace or flat) of $V$ is a subset of the form $$ L = v_0 + W = {, v_0 + w : w \in W ,}, $$ where $W \subseteq V$ is a linear subspace, called the direction (or direction space) of $L$ , and $v_{0} \in V$ is a base point. The empty set is included by convention only where stated; otherwise "linear manifold" means a nonempty coset of a subspace. The dimension of $L$ is $dim L := dim W$ . A flat of dimension $0$ is a point, of dimension $1$ a line, of dimension $2$ a plane, and of dimension $dim V - 1$ a hyperplane.

The base point is not unique, but the direction is. If $L = v_{0} + W = v_{1} + W^{'}$ as sets, then $W = W^{'} = L - L := {x - y : x, y \in L}$ , and $v_{1} \in L$ is an admissible base point precisely when $v_{1} - v_{0} \in W$ . Thus $L$ determines $W$ exactly, and determines $v_{0}$ only modulo $W$ .

Definition (affine combination). An affine combination of points $x_{0}, \dots, x_{k} \in V$ is a linear combination $\sum_{i = 0}^{k} λ_{i} x_{i}$ whose coefficients satisfy the affine constraint $\sum_{i = 0}^{k} λ_{i} = 1$ . A set $S \subseteq V$ is affinely closed when every affine combination of finitely many points of $S$ again lies in $S$ .

Definition (affine independence and affine span). Points $p_{0}, \dots, p_{k} \in V$ are affinely independent when the displacement vectors $p_{1} - p_{0}, \dots, p_{k} - p_{0}$ are linearly independent in $V$ ; this condition does not depend on which point is singled out as $p_{0}$ . The affine span $aff {p_{0}, \dots, p_{k}}$ is the smallest linear manifold containing them, namely $p_{0} + span {p_{1} - p_{0}, \dots, p_{k} - p_{0}}$ . When the points are affinely independent, $dim aff {p_{0}, \dots, p_{k}} = k$ , and every point $x$ of the span has a unique tuple of barycentric coordinates $(λ_{0}, \dots, λ_{k})$ with $\sum_{i} λ_{i} = 1$ and $x = \sum_{i} λ_{i} p_{i}$ .

Definition (hyperplane). A hyperplane of $V$ is a linear manifold $H$ whose direction has codimension $1$ , that is $dim (V / W) = 1$ where $W$ is the direction of $H$ . In finite dimension $n$ this is $dim H = n - 1$ .

Definition (parallelism, half-spaces). Two linear manifolds $L, M$ are parallel when the direction of one contains the direction of the other, $W_{L} \subseteq W_{M}$ or $W_{M} \subseteq W_{L}$ . Over $K = R$ , a hyperplane $H = {x : f (x) = c}$ with $f \in V^{*}$ nonzero determines two closed half-spaces $H^{+} = {x : f (x) \geq c}$ and $H^{-} = {x : f (x) \leq c}$ , with $H = H^{+} \cap H^{-}$ .

Notation: $L = v_{0} + W$ is the coset of $W$ through $v_{0}$ ; $dim L = dim W$ ; $L - L$ is the difference set, equal to the direction; $aff S$ is the affine span of $S$ ; $f^{- 1} (c) = {x : f (x) = c}$ is the level set of $f \in V^{*}$ at $c$ ; $V / W$ is the quotient space, whose dimension is the codimension of $W$ .

Counterexamples to common slips

A linear manifold is not a subspace unless it contains $0$ . The set ${(x, y) : x + y = 1}$ is a line in $R^{2}$ but is not closed under addition: $(1, 0)$ and $(0, 1)$ lie on it, yet their sum $(1, 1)$ does not. It is closed under affine combinations, not linear ones.
The base point is not part of the data. Writing $L = v_{0} + W$ tempts one to treat $v_{0}$ as intrinsic, but any other point of $L$ serves equally; only the difference set $L - L = W$ is forced by $L$ .
"Hyperplane" means codimension one, not dimension one. In $R^{4}$ a hyperplane is three-dimensional. The defining feature is a single linear equation, not a single direction.
The defining functional of a hyperplane is unique only up to a common nonzero scalar. The equations $x + y = 1$ and $2 x + 2 y = 2$ cut out the same line; $(f, c)$ and $(α f, α c)$ describe one hyperplane for every nonzero $α$ .

Key theorem with proof Intermediate+

Theorem (three faces of a linear manifold; Shilov Ch. 2 ^{[source pending]}; Berger Ch. 2 ^{[source pending]}). Let $V$ be a $K$ -vector space and $L \subseteq V$ a nonempty subset. The following are equivalent.

$L$ is a coset $L = v_{0} + W$ of a subspace $W$ .
$L$ is affinely closed: every affine combination $\sum_{i} λ_{i} x_{i}$ of points of $L$ with $\sum_{i} λ_{i} = 1$ lies in $L$ .
For some, equivalently every, point $v_{0} \in L$ , the difference set $L - v_{0} = {x - v_{0} : x \in L}$ is a subspace of $V$ .

Moreover, $L$ is a hyperplane if and only if $L = f^{- 1} (c) = {x : f (x) = c}$ for some nonzero linear functional $f \in V^ $an d so m esc a l a r$ c \in K $; an d t h e p ai r$ (f, c) $i s d e t er min e d b y$ L$ up to a common nonzero scalar factor.*

Proof. $(1) \Rightarrow (2)$ . Suppose $L = v_{0} + W$ with $W$ a subspace. Take points $x_{i} = v_{0} + w_{i}$ of $L$ , with $w_{i} \in W$ , and scalars $λ_{i}$ summing to $1$ . Then $$ \sum_i \lambda_i x_i = \sum_i \lambda_i (v_0 + w_i) = \Big(\sum_i \lambda_i\Big) v_0 + \sum_i \lambda_i w_i = v_0 + \sum_i \lambda_i w_i, $$ using $\sum_{i} λ_{i} = 1$ in the last step. Since $W$ is a subspace, $\sum_{i} λ_{i} w_{i} \in W$ , so the affine combination equals $v_{0} + (element of W) \in L$ . Thus $L$ is affinely closed.

$(2) \Rightarrow (3)$ . Assume $L$ is affinely closed and fix any $v_{0} \in L$ ; set $W := L - v_{0}$ . To show $W$ is a subspace we check it contains $0$ and is closed under sums and scalar multiples. First $0 = v_{0} - v_{0} \in W$ . For closure under scaling, let $w = x - v_{0} \in W$ with $x \in L$ and let $μ \in K$ . The combination $μx + (1 - μ) v_{0}$ is an affine combination of $x, v_{0} \in L$ (coefficients sum to $1$ ), so it lies in $L$ ; subtracting $v_{0}$ , $$ \big(\mu x + (1-\mu) v_0\big) - v_0 = \mu (x - v_0) = \mu w \in W. $$ For closure under addition, let $w_{1} = x_{1} - v_{0}$ and $w_{2} = x_{2} - v_{0}$ with $x_{1}, x_{2} \in L$ . The point $x_{1} + x_{2} - v_{0}$ is the affine combination $1 \cdot x_{1} + 1 \cdot x_{2} + (- 1) \cdot v_{0}$ , whose coefficients sum to $1$ , so it lies in $L$ ; subtracting $v_{0}$ , $$ (x_1 + x_2 - v_0) - v_0 = (x_1 - v_0) + (x_2 - v_0) = w_1 + w_2 \in W. $$ Hence $W$ is a subspace. The same construction with any other base point $v_{1} \in L$ yields $L - v_{1}$ , which differs from $L - v_{0}$ by the translation $v_{0} - v_{1} \in W$ and is therefore the same subspace; so the condition holds for every point once it holds for one.

$(3) \Rightarrow (1)$ . If $W = L - v_{0}$ is a subspace, then $L = v_{0} + W$ by construction, which is statement $(1)$ .

Hyperplane correspondence. Suppose first $L = f^{- 1} (c)$ with $f \in V^{*}$ nonzero. Since $f \neq = 0$ there is $a$ with $f (a) = 1$ ; then $v_{0} := c a$ satisfies $f (v_{0}) = c$ , so $v_{0} \in L$ and $L$ is nonempty. For $x \in V$ , $f (x) = c$ if and only if $f (x - v_{0}) = 0$ , that is $x - v_{0} \in ker f$ . Hence $L = v_{0} + ker f$ , a coset of the subspace $ker f$ . By the rank-nullity theorem for the functional $f : V \to K$ from 01.01.05, $im f = K$ has dimension $1$ , so $V / ker f ≅ im f = K$ has dimension $1$ ; thus $ker f$ has codimension $1$ and $L$ is a hyperplane.

Conversely, let $L = v_{0} + W$ be a hyperplane, so $dim (V / W) = 1$ . The quotient map $π : V \to V / W$ followed by any linear isomorphism $V / W ≅ K$ gives a nonzero functional $g \in V^{*}$ with $ker g = W$ . Set $c := g (v_{0})$ and $f := g$ . For $x \in V$ , $x \in L$ means $x - v_{0} \in W = ker g$ , that is $g (x) = g (v_{0}) = c$ ; so $L = f^{- 1} (c)$ .

Uniqueness up to scalar. Suppose $f^{- 1} (c) = g^{- 1} (d) = L$ with $f, g$ nonzero. Both kernels equal the direction $W$ of $L$ , so $ker f = ker g$ is a codimension-one subspace. Two functionals with the same codimension-one kernel are proportional: choose $a$ with $f (a) \neq = 0$ ; every $x \in V$ decomposes as $x = \frac{f ( x )}{f ( a )} a + w$ with $w \in ker f = ker g$ , whence $g (x) = \frac{f ( x )}{f ( a )} g (a)$ , so $g = α f$ with $α = g (a) / f (a) \neq = 0$ . Evaluating at any $v_{0} \in L$ gives $d = g (v_{0}) = α f (v_{0}) = α c$ . Thus $(g, d) = α (f, c)$ . $□$

Bridge. The coset model $L = v_{0} + W$ builds toward the quotient construction $V / W$ of 01.01.04: the points of the quotient are the parallel cosets, so the set of all flats with a fixed direction $W$ is exactly the vector space $V / W$ , and the dimension count $dim L = dim W$ is the source-side reading of $dim V = dim W + dim (V / W)$ . The hyperplane-as-level-set correspondence appears again in the dual-space pairing of 01.01.02, where a hyperplane through the origin in $V$ is precisely a point of the projectivised dual $P (V^{*})$ , the seed of the points-versus-hyperplanes duality of projective geometry. The affine-combination characterisation connects forward to convex geometry, where restricting the coefficients to $λ_{i} \geq 0$ turns affine spans into convex hulls and hyperplanes into the separating hyperplanes of the Hahn-Banach circle. And the solution-set reading $x_{p} + ker A$ specialises here the Kronecker-Capelli theorem of 01.01.06: every consistent linear system presents its solution flat as the intersection of the hyperplanes cut out by its individual equations.

Exercises Intermediate+

Exercise 4 (medium, symbolic).

Consider the system over $R$ $$ x_1 + x_2 + x_3 + x_4 = 2, \qquad x_1 - x_2 + x_3 - x_4 = 0. $$ Exhibit its solution set as a linear manifold $x_{p} + ker A$ : give a particular solution $x_{p}$ and a basis of the direction space, and state the dimension.

Hint

Add and subtract the equations to get $x_{1} + x_{3} = 1$ and $x_{2} + x_{4} = 1$ . Choose two free variables.

Answer

Adding the equations gives $2 (x_{1} + x_{3}) = 2$ , so $x_{1} + x_{3} = 1$ ; subtracting gives $2 (x_{2} + x_{4}) = 2$ , so $x_{2} + x_{4} = 1$ . Take $x_{3} = s$ and $x_{4} = t$ free; then $x_{1} = 1 - s$ and $x_{2} = 1 - t$ . A particular solution is $x_{p} = (1, 1, 0, 0)$ (set $s = t = 0$ ). The direction space is $$ \ker A = \operatorname{span}{(-1, 0, 1, 0),\ (0, -1, 0, 1)}, $$ the two basis vectors arising from $s$ and $t$ . The solution flat is $x_{p} + ker A$ , of dimension $2$ , and indeed $dim ker A = 4 - rank (A) = 4 - 2 = 2$ , consistent with 01.01.06. Rubric: full credit for $x_{p}$ , the two-vector kernel basis, and dimension $2$ .

Exercise 6 (medium, proof).

Prove that two distinct parallel hyperplanes in $V$ are disjoint, and conversely that two disjoint hyperplanes are parallel.

Hint

Parallel hyperplanes share a direction $W$ . Use the level-set form $f (x) = c$ and $f (x) = d$ with the same $f$ .

Answer

Write the hyperplanes as $H_{1} = f^{- 1} (c)$ and $H_{2} = g^{- 1} (d)$ with nonzero functionals $f, g$ . Parallel means equal directions $ker f = ker g$ , which forces $g = α f$ for a nonzero scalar $α$ , so we may rescale and take $g = f$ . Then $H_{1} = f^{- 1} (c)$ and $H_{2} = f^{- 1} (d)$ . If $x \in H_{1} \cap H_{2}$ then $c = f (x) = d$ , so the hyperplanes coincide; being distinct, $c \neq = d$ , and the intersection is empty — they are disjoint. Conversely, suppose $H_{1} \cap H_{2} = \emptyset$ . If the directions differed, $ker f \neq = ker g$ , then $ker f + ker g = V$ (two distinct codimension-one subspaces span the whole space), and one checks the affine map $x \mapsto (f (x), g (x))$ is then surjective onto $K^{2}$ , so the value $(c, d)$ is attained, giving a common point — contradicting disjointness. Hence the directions agree and the hyperplanes are parallel. Rubric: full credit for both directions, including the surjectivity argument for the converse.

Exercise 7 (hard, proof).

Let $L = v_{0} + W$ be a $k$ -dimensional flat in $V$ with $dim V = n$ . Prove that $L$ is the intersection of $n - k$ hyperplanes, and that no fewer suffice.

Hint

The direction $W = ker f_{1} \cap \dots \cap ker f_{m}$ for functionals $f_{i}$ iff ${f_{i}}$ spans the annihilator $W^{\circ} \subseteq V^{*}$ . Use $dim W^{\circ} = n - k$ .

Answer

Let $W^{\circ} = {f \in V^{*} : f ∣_{W} = 0}$ be the annihilator of $W$ ; from 01.01.02, $dim W^{\circ} = n - dim W = n - k$ . Choose a basis $f_{1}, \dots, f_{n - k}$ of $W^{\circ}$ . Then $W = ⋂_{i} ker f_{i}$ : indeed $W \subseteq ker f_{i}$ for each $i$ by definition of the annihilator, and conversely $⋂_{i} ker f_{i}$ has dimension $n - rank [f_{1}, \dots, f_{n - k}] = n - (n - k) = k = dim W$ , forcing equality. Setting $c_{i} := f_{i} (v_{0})$ , the hyperplanes $H_{i} = f_{i}^{- 1} (c_{i})$ satisfy $⋂_{i} H_{i} = v_{0} + ⋂_{i} ker f_{i} = v_{0} + W = L$ . So $L$ is an intersection of $n - k$ hyperplanes. For the lower bound: each hyperplane constraint can cut the dimension by at most one, so an intersection of $m$ hyperplanes has dimension at least $n - m$ ; to reach dimension $k$ one needs $n - m \leq k$ , that is $m \geq n - k$ . Hence $n - k$ is the minimum. Rubric: full credit for the annihilator-basis construction and the dimension-counting lower bound.

Exercise 8 (hard, symbolic).

Find the equation of the hyperplane in $R^{3}$ through the three points $a = (1, 0, 0)$ , $b = (0, 2, 0)$ , $c = (0, 0, 3)$ , and verify each point satisfies it.

Hint

Seek $f (x, y, z) = α x + β y + γ z = c$ holding at all three points; or use the normal $n = (b - a) \times (c - a)$ .

Answer

A nonzero functional $f (x, y, z) = α x + β y + γ z$ takes the same value $c$ at $a, b, c$ . From $a$ : $α = c$ . From $b$ : $2 β = c$ , so $β = c /2$ . From $c$ : $3 γ = c$ , so $γ = c /3$ . Choosing $c = 6$ clears denominators: $α = 6$ , $β = 3$ , $γ = 2$ , giving the equation $$ 6x + 3y + 2z = 6, \qquad \text{equivalently} \qquad \frac{x}{1} + \frac{y}{2} + \frac{z}{3} = 1. $$ Check: at $a$ , $6 (1) + 3 (0) + 2 (0) = 6$ ; at $b$ , $6 (0) + 3 (2) + 2 (0) = 6$ ; at $c$ , $6 (0) + 3 (0) + 2 (3) = 6$ . All three satisfy it. The intercept form $\frac{x}{1} + \frac{y}{2} + \frac{z}{3} = 1$ reads off the axis crossings. Rubric: full credit for $6 x + 3 y + 2 z = 6$ (or any nonzero scalar multiple) and the three checks.

Advanced results Master

Theorem (the affine group and the structure of affine maps; Berger Ch. 2 ^{[source pending]}; Gallier Ch. 2 ^{[source pending]}). Let $V$ be a finite-dimensional $K$ -vector space. A map $φ : V \to V$ is an affine map — one preserving affine combinations, $φ (\sum_{i} λ_{i} x_{i}) = \sum_{i} λ_{i} φ (x_{i})$ whenever $\sum_{i} λ_{i} = 1$ — if and only if $φ (x) = A x + b$ for a unique linear map $A \in End (V)$ , the linear part, and a unique vector $b \in V$ . The invertible affine maps form the affine group $$ \operatorname{Aff}(V) = V \rtimes \operatorname{GL}(V), $$ the semidirect product of the translation group $V$ by the general linear group, with multiplication $(b_{1}, A_{1}) (b_{2}, A_{2}) = (b_{1} + A_{1} b_{2}, A_{1} A_{2})$ . The linear part is the homomorphism $Aff (V) \to GL (V)$ , $(b, A) \mapsto A$ , whose kernel is the translation subgroup. Affine maps send flats to flats, preserve dimension when invertible, preserve parallelism, and preserve barycentric coordinates — these last being the complete affine invariants, in the sense that ratios of collinear lengths are preserved while absolute lengths and angles are not. The affine group sits inside the projective group of the projective completion as the stabiliser of the hyperplane at infinity, which is the precise statement that affine geometry is projective geometry with one hyperplane distinguished.

Theorem (projective completion and points at infinity). Embed $V$ as the affine hyperplane ${(v, 1) : v \in V} \subset V \oplus K$ . The projective space $P (V \oplus K)$ — the set of lines through the origin of $V \oplus K$ — decomposes as $$ \mathbb{P}(V \oplus K) = \underbrace{V}{\text{affine part}} \ \sqcup\ \underbrace{\mathbb{P}(V)}{\text{hyperplane at infinity}}, $$ where the affine part is the image of the embedding and the hyperplane at infinity $P (V)$ is the set of directions: each line of $V$ acquires exactly one point at infinity, namely its direction, and two affine lines meet at infinity precisely when they are parallel. In this completion a flat $L = v_{0} + W$ of $V$ closes up to a projective subspace $\overline{L}$ whose points at infinity are $P (W)$ , the directions of $L$ ; parallelism becomes incidence at infinity, and the troublesome case-splitting of affine intersection — meet, or be parallel — collapses into the uniform projective statement that any two projective subspaces meet in the expected dimension.

Theorem (points–hyperplanes duality). In $P (V)$ with $dim V = n + 1$ , the hyperplanes are exactly the projective subspaces of dimension $n - 1$ , and the map sending a hyperplane ${[x] : f (x) = 0}$ to the line $[f] \in \mathbb{P}(V^)$ is a bijection* $$ {\text{hyperplanes of } \mathbb{P}(V)} \ \xrightarrow{\ \sim\ }\ \mathbb{P}(V^). $$ This bijection is a duality: it inverts incidence, sending the pencil of hyperplanes through a fixed point to the points of a fixed hyperplane in $\mathbb{P}(V^) $, an d i d e n t i f i es t h e d o u b l e d u a l$ \mathbb{P}(V^{**}) $w i t h$ \mathbb{P}(V)$ through the canonical evaluation of 01.01.02. Every theorem about points and lines in the plane thereby acquires a dual theorem with the words "point" and "line" interchanged — the symmetry that makes Desargues' theorem self-dual and pairs Pascal's theorem with Brianchon's.

Synthesis. A flat carries two separable pieces of data, a direction subspace $W$ and a position, and every elementary operation on flats acts on these two pieces independently. Dimension, parallelism, and the count of defining equations are properties of the direction alone, read in the quotient $V / W$ and its annihilator $W^{\circ} \subseteq V^{*}$ ; position enters only through a single base point modulo $W$ , which is why the set of flats of fixed direction is itself the vector space $V / W$ . The coset model and the affine-combination model are two presentations of this same separation: the coset displays the direction explicitly, the affine combination encodes it implicitly through the constraint $\sum λ_{i} = 1$ that quotients out the position. The hyperplane is the boundary case codimension one, where the direction is the kernel of a single functional and the position is the single scalar value of that functional; this is what makes a linear system a list of hyperplanes and its solution set their common flat, the geometric content of Kronecker-Capelli 01.01.06.

Passing to the projective completion absorbs the position data into the geometry by adjoining the directions as honest points at infinity, so that the affine group is recovered as the stabiliser of the hyperplane at infinity and the affine case-splitting of intersection dissolves into uniform projective incidence. Duality then exchanges the two pieces one level up: a hyperplane in $P (V)$ is a point in $P (V^{*})$ , the annihilator of a flat is the flat of its defining functionals, and the entire affine theory of flats becomes a shadow, on a distinguished hyperplane, of the symmetric projective theory of subspaces.

Full proof set Master

Proposition (the difference set is the direction, and the base point is determined modulo it). Let $L = v_{0} + W$ be a nonempty linear manifold. Then $L - L := {x - y : x, y \in L} = W$ , and a vector $v_{1}$ is an admissible base point — meaning $L = v_{1} + W$ — if and only if $v_{1} \in L$ .

Proof. Take $x, y \in L$ , so $x = v_{0} + w_{1}$ and $y = v_{0} + w_{2}$ with $w_{1}, w_{2} \in W$ ; then $x - y = w_{1} - w_{2} \in W$ , giving $L - L \subseteq W$ . Conversely, for $w \in W$ , the points $x = v_{0} + w$ and $y = v_{0}$ lie in $L$ and have difference $x - y = w$ , so $W \subseteq L - L$ . Hence $L - L = W$ , an invariant of $L$ independent of the base point. For the base-point claim: if $v_{1} \in L$ then $v_{1} = v_{0} + w_{*}$ for some $w_{*} \in W$ , and $v_{1} + W = v_{0} + w_{*} + W = v_{0} + W = L$ since $w_{*} + W = W$ . Conversely, if $L = v_{1} + W$ then $v_{1} = v_{1} + 0 \in L$ . $□$

Proposition (affine span and barycentric coordinates are well-defined). Let $p_{0}, \dots, p_{k} \in V$ be affinely independent. Then $aff {p_{0}, \dots, p_{k}} = {\sum_{i} λ_{i} p_{i} : \sum_{i} λ_{i} = 1}$ has dimension $k$ , and each of its points has a unique tuple of barycentric coordinates $(λ_{0}, \dots, λ_{k})$ .

Proof. Write $u_{i} := p_{i} - p_{0}$ for $i = 1, \dots, k$ ; affine independence means $u_{1}, \dots, u_{k}$ are linearly independent, so they span a subspace $W$ of dimension $k$ . A point $x = \sum_{i = 0}^{k} λ_{i} p_{i}$ with $\sum_{i} λ_{i} = 1$ rewrites, using $λ_{0} = 1 - \sum_{i \geq 1} λ_{i}$ , as $$ x = p_0 + \sum_{i=1}^k \lambda_i (p_i - p_0) = p_0 + \sum_{i=1}^k \lambda_i u_i \in p_0 + W, $$ and conversely every element of $p_{0} + W$ arises this way. Hence the set of affine combinations equals $p_{0} + W$ , a flat of dimension $dim W = k$ , and it is the smallest flat containing the $p_{i}$ because any flat containing them contains $p_{0}$ and all displacements $u_{i}$ , hence $p_{0} + W$ . For uniqueness, suppose $\sum_{i} λ_{i} p_{i} = \sum_{i} μ_{i} p_{i}$ with both coefficient tuples summing to $1$ . Subtracting and grouping at $p_{0}$ , $$ 0 = \sum_{i=1}^k (\lambda_i - \mu_i) u_i, $$ and linear independence of the $u_{i}$ forces $λ_{i} = μ_{i}$ for $i \geq 1$ ; the constraint $\sum_{i} λ_{i} = \sum_{i} μ_{i} = 1$ then forces $λ_{0} = μ_{0}$ . The barycentric coordinates are therefore unique. $□$

Proposition (intersection of a finite family of flats is a flat or empty; the parallel dichotomy). Let ${L_{j}}_{j \in J}$ be a finite family of linear manifolds with directions $W_{j}$ . If $⋂_{j} L_{j} \neq = \emptyset$ , then it is a linear manifold with direction $⋂_{j} W_{j}$ ; in particular two flats $L = p + W$ , $M = q + U$ either meet, in which case $L \cap M = r + (W \cap U)$ for any common point $r$ , or are disjoint, and disjoint flats with $W \subseteq U$ or $U \subseteq W$ are parallel.

Proof. Suppose $p \in ⋂_{j} L_{j}$ , so $L_{j} = p + W_{j}$ for every $j$ by the base-point proposition. A point $x$ lies in $⋂_{j} L_{j}$ if and only if $x - p \in W_{j}$ for every $j$ , that is $x - p \in ⋂_{j} W_{j}$ . Thus $⋂_{j} L_{j} = p + ⋂_{j} W_{j}$ , and $⋂_{j} W_{j}$ is a subspace as an intersection of subspaces; so the intersection is a flat of direction $⋂_{j} W_{j}$ . The two-flat statement is the case $J = {1, 2}$ . For the dichotomy: when $L \cap M = \emptyset$ the intersection is by definition empty; the additional remark records that if moreover one direction contains the other the flats are parallel by definition, which is the disjoint configuration of nested directions — for instance two distinct parallel lines, where $W = U$ and the difference $q - p \in / W$ obstructs a common point. $□$

Proposition (a hyperplane separates its complement into two pieces over an ordered field). Let $K$ be an ordered field, $V$ a $K$ -vector space, and $H = f^{- 1} (c)$ a hyperplane with $f \in V^ $n o n z er o . T h eco m pl e m e n t$ V \setminus H $i s t h e d i s j o in t u ni o n o f t h e tw oo p e nha l f - s p a ces$ H^{+}{\circ} = {x : f(x) > c} $an d$ H^{-}{\circ} = {x : f(x) < c} $, e a c hi sco n v e x, an d an y se g m e n t$ [x, y] = {(1-t)x + ty : 0 \le t \le 1} $j o inin g a p o in t o f$ H^{+}{\circ} $t o a p o in t o f$ H^{-}{\circ} $m ee t s$ H$ in exactly one point.*

Proof. For $x \in / H$ , $f (x) \neq = c$ , so $f (x) > c$ or $f (x) < c$ but not both, by trichotomy in the ordered field $K$ ; this partitions $V ∖ H$ into $H_{\circ}^{+}$ and $H_{\circ}^{-}$ disjointly. Convexity of $H_{\circ}^{+}$ : if $f (x) > c$ and $f (y) > c$ and $0 \leq t \leq 1$ , then $f ((1 - t) x + t y) = (1 - t) f (x) + t f (y) > (1 - t) c + t c = c$ , using that a convex combination of two quantities each exceeding $c$ exceeds $c$ ; likewise for $H_{\circ}^{-}$ . For the segment crossing, parametrise $g (t) := f ((1 - t) x + t y) = (1 - t) f (x) + t f (y)$ , an affine function of $t$ with $g (0) = f (x) > c$ and $g (1) = f (y) < c$ . Solving $g (t) = c$ gives the unique $$ t_* = \frac{f(x) - c}{f(x) - f(y)} \in (0, 1), $$ the denominator nonzero since $f (x) \neq = f (y)$ , and $t_{*} \in (0, 1)$ because numerator and denominator are both positive and the numerator is the smaller. The single crossing point $(1 - t_{*}) x + t_{*} y$ lies on $H$ , and no other value of $t$ solves the affine equation $g (t) = c$ . $□$

Connections Master

The coset model $L = v_{0} + W$ is the geometric face of the quotient space $V / W$ of 01.01.04: the points of the quotient are exactly the parallel cosets of $W$ , so the family of all flats sharing a direction $W$ is itself a vector space, and the rank-nullity identity $dim V = dim W + codim W$ is the dimension count $dim L + dim (V / W) = dim V$ read on flats.

The hyperplane-as-level-set correspondence is dual to the dual-space theory of 01.01.02: a hyperplane through the origin is the kernel of a nonzero functional, the annihilator $W^{\circ}$ of a $k$ -flat's direction is an $(n - k)$ -dimensional space of defining functionals, and the points-versus-hyperplanes duality of projective space is the projectivisation of the canonical pairing $V \times V^{*} \to K$ . The same annihilator computation reappears in the four-fundamental-subspaces orthogonality of 01.01.10, where over an inner-product space the defining functional is realised by the normal vector and the half-spaces acquire a metric meaning.

The affine structure of the solution set $x_{p} + ker A$ is precisely the Kronecker-Capelli theorem of 01.01.06 read geometrically: a consistent linear system is an intersection of hyperplanes, its solution flat has direction $ker A$ and dimension $n - rank A$ , and the particular-plus-homogeneous decomposition is the choice of a base point plus the direction. This flat-of-solutions picture propagates to the geodesics and affine connections of 13.02.01, where the flat affine structure of $R^{n}$ is the local model that a connection curves, and the affine group reappears as the structure group of an affine bundle.

Historical & philosophical context Master

The systematic idea that geometry could be done with points described by weights, rather than by coordinates relative to an origin, is due to August Ferdinand Möbius, whose 1827 Der barycentrische Calcul introduced barycentric coordinates: a point of a triangle or simplex specified by the masses one would place at its vertices to balance there ^{[Möbius 1827]}. The barycentric description is intrinsically affine — it never names an origin — and it is the historical source of the affine-combination characterisation of flats used in this unit. The general $n$ -dimensional theory of linear extension, in which flats are spanned by points and carry a dimension independent of any fixed coordinate frame, was constructed by Hermann Grassmann in the 1844 Ausdehnungslehre ^{[Grassmann 1844]}. The separation of the affine from the projective and the metric, and the recognition that affine geometry is projective geometry with a distinguished hyperplane at infinity, belongs to the nineteenth-century projective school and was given its group-theoretic form in Felix Klein's Erlangen programme.

The treatment of linear manifolds as cosets of subspaces, with hyperplanes as level surfaces of a linear form and the solution set of a system as a manifold, is the form in Georgi Shilov's Linear Algebra (1971 English translation) followed here; the modern axiomatic affine-space framework, in which the difference of two points is a vector and the affine group is the semidirect product $V ⋊ GL (V)$ , is the presentation of Marcel Berger's Geometry I (1987) and of Jean Gallier's Geometric Methods ^{[Berger 1987]}.

Bibliography Master

@book{Mobius1827,
  author    = {M\"obius, August Ferdinand},
  title     = {Der barycentrische Calcul},
  publisher = {Johann Ambrosius Barth},
  address   = {Leipzig},
  year      = {1827}
}

@book{Grassmann1844,
  author    = {Grassmann, Hermann},
  title     = {Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik},
  publisher = {Otto Wigand},
  address   = {Leipzig},
  year      = {1844}
}

@book{Shilov1977,
  author    = {Shilov, Georgi E.},
  title     = {Linear Algebra},
  publisher = {Dover Publications},
  address   = {New York},
  year      = {1977},
  note      = {Translation of the 1971 Russian edition, transl. R. A. Silverman}
}

@book{Berger1987,
  author    = {Berger, Marcel},
  title     = {Geometry I},
  series    = {Universitext},
  publisher = {Springer-Verlag},
  address   = {Berlin},
  year      = {1987},
  note      = {Translation of G\'eom\'etrie, Cedic/Nathan, 1977}
}

@book{Gallier2011,
  author    = {Gallier, Jean},
  title     = {Geometric Methods and Applications: For Computer Science and Engineering},
  edition   = {2nd},
  series    = {Texts in Applied Mathematics},
  volume    = {38},
  publisher = {Springer},
  year      = {2011}
}

@book{Artin1957,
  author    = {Artin, Emil},
  title     = {Geometric Algebra},
  publisher = {Interscience Publishers},
  address   = {New York},
  year      = {1957}
}

@book{Bourbaki1970,
  author    = {Bourbaki, Nicolas},
  title     = {Alg\`ebre, Chapitres 1 \`a 3},
  publisher = {Hermann},
  address   = {Paris},
  year      = {1970}
}

Prerequisites

01.01.04
01.01.06
01.01.02

Tier anchors

beginner: A line or a plane that need not pass through the origin — shift a subspace by a fixed offset and you get a flat. Shilov *Linear Algebra* Ch. 2; 3Blue1Brown *Essence of Linear Algebra* Ch. 3 (linear combinations, span)
intermediate: Shilov *Linear Algebra* Ch. 2 (linear manifolds as cosets, hyperplanes as level sets of a linear functional); Hoffman-Kunze *Linear Algebra* §3.5 (the transpose and the dual hyperplane); Lang *Linear Algebra* Ch. I (cosets and affine subspaces)
master: Shilov *Linear Algebra* Ch. 2; Berger *Geometry I* Ch. 2–3 (affine spaces, affine maps, the affine group, the projective completion); Gallier *Geometric Methods and Applications* Ch. 2–3 (affine and projective geometry); Bourbaki *Algèbre* Ch. II §9 (affine and projective spaces); Artin *Geometric Algebra* Ch. II

References

images/Shilov-Linear-Algebra__4cbdee00cc.jpg · Shilov *Linear Algebra* — Fast Track archive cover; Ch. 2 the linear manifold $L = x_0 + L'$ as a coset of a subspace $L'$, its dimension, the hyperplane as the level set of a linear form, and the solution set of a linear system as a linear manifold
Shilov, G. E. — Linear Algebra (Dover, 1977 transl. of the 1971 Russian ed.) · Ch. 2 — linear manifolds (cosets of subspaces), dimension of a manifold, hyperplanes as level surfaces of a linear form, the geometry of the solution set of a linear system
Berger, M. — Geometry I (Universitext, Springer, 1987; transl. of Géométrie, Cedic/Nathan 1977) · Ch. 2–3 — affine spaces, affine subspaces and their directions, affine combinations and barycentres, the affine group, parallelism, and the projective completion
Gallier, J. — Geometric Methods and Applications: For Computer Science and Engineering (2nd ed., Springer, 2011) · Ch. 2–3 — affine spaces, affine maps, barycentric coordinates, affine frames, half-spaces and separating hyperplanes, the projective completion of an affine space
Möbius, A. F. — Der barycentrische Calcul · Johann Ambrosius Barth, Leipzig, 1827 — the barycentric calculus: a point of a flat described by mass-weights at the vertices of an affinely independent frame, the origin of barycentric coordinates
Grassmann, H. — Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik · Otto Wigand, Leipzig, 1844 — the first systematic $n$-dimensional theory of linear extension, of flats spanned by points, and of their dimension independent of a fixed origin

Estimated time

beginner: 18m
intermediate: 44m
master: 82m