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A subspace is the way linear algebra recognises a smaller linear world inside a larger one. A subset of a [vector space](/page/Vector%20Space) may contain many vectors, but it supports the same linear methods only when it is stable under the operations that make vectors useful: addition and scalar multiplication. Once that stability is present, the subset inherits bases, dimension, [linear maps](/page/Linear%20Map), quotient constructions, kernels, images, and direct-sum decompositions from the ambient theory.

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The point of the definition is not merely to name certain subsets. It separates subsets on which linear equations still make sense from subsets that are only set-theoretic pieces of a vector space. Lines through the origin in $\mathbb{R}^2$ are subspaces; affine lines not passing through the origin are not. The difference is structural: a subspace contains the zero vector and is closed under every linear combination of its elements.

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## Definition

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The ambient object is a vector space over a field. The subset deserves to be called a subspace exactly when the vector space operations do not leave the subset. This definition is the local test that tells us whether a subset can be treated as a vector space in its own right.

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[definition: Subspace] Let $F$ be a field, and let $V$ be an $F$-vector space. A subset $U \subset V$ is a subspace of $V$ if the following conditions hold: 1. $0_V \in U$. 2. If $u, v \in U$, then $u + v \in U$. 3. If $u \in U$ and $\lambda \in F$, then $\lambda u \in U$. [/definition]

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A subspace is therefore not additional structure placed on $U$; it is a condition saying that the structure already on $V$ restricts to $U$. In calculations, however, checking addition and scalar multiplication separately can obscure the real invariant: closure under linear combinations. The following definition isolates that invariant so that subspace verification can match the way vectors are actually manipulated.

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[definition: Linear Combination Closure] Let $F$ be a field, let $V$ be an $F$-vector space, and let $U \subset V$. The subset $U$ is closed under binary linear combinations if for all $u, v \in U$ and all $\lambda, \mu \in F$, one has \begin{align*} \lambda u + \mu v \in U. \end{align*} [/definition]

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Linear combination closure packages the two closure axioms into one condition, but it says nothing by itself if the set is empty. In the vector-space setting, the resulting working rule is the standard subspace test: a nonempty subset $U \subset V$ is a subspace exactly when it is closed under all binary linear combinations $\lambda u + \mu v$ with $u,v \in U$ and $\lambda,\mu \in F$.

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This rule is useful because it matches ordinary calculations. Instead of verifying addition and scalar multiplication as separate tasks, one checks that every linear combination of two already-admissible vectors remains admissible. The nonempty hypothesis is part of the test, not a cosmetic add-on: without it, the empty set would satisfy the closure condition vacuously while still failing to contain the zero vector required of a subspace.

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## Equivalent Characterisations

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Subspaces are also exactly the subsets that can be described as all linear combinations of some family of vectors. This point of view explains why subspaces are the natural domains for [basis](/page/Basis) and [dimension](/page/Dimension). Before using generators to describe subspaces, we need a construction that turns any proposed family of vectors into the collection of all vectors it can produce.

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[definition: Span] Let $F$ be a field, let $V$ be an $F$-vector space, and let $S \subset V$. The span of $S$ is the subset \begin{align*} \operatorname{span}(S) = \left\{\sum_{i=1}^{n} \lambda_i s_i : n \in \mathbb{N},\ s_i \in S,\ \lambda_i \in F\right\}. \end{align*} If $S = \varnothing$, then $\operatorname{span}(S) = \{0_V\}$. [/definition]

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The span construction starts with an arbitrary set and forces closure under finite linear combinations. For it to be useful, it must be the least possible subspace containing the original set, rather than merely some large subspace built from it. Thus $\operatorname{span}(S)$ is the subspace generated by $S$: it contains $S$, it is closed under finite linear combinations, and every subspace of $V$ that contains $S$ must also contain $\operatorname{span}(S)$.

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This minimality is what makes spanning sets precise. It says that a vector lies in $\operatorname{span}(S)$ only because it can be built from finitely many vectors of $S$ using the vector-space operations; no extra vectors are included for unrelated reasons. In the language used later for bases, $S$ spans a subspace exactly when its finite linear combinations account for the whole subspace.

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Generators describe subspaces from inside, while maps produce subspaces by testing or transporting vectors. Since linear maps preserve linear combinations, the vectors sent to zero should form a closed linear system inside the domain. The following definition formalises this solution space for homogeneous linear equations.

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[definition: Kernel] Let $F$ be a field, and let $T: V \to W$ be a linear map of $F$-vector spaces. The kernel of $T$ is \begin{align*} \ker(T) = \{v \in V : T(v) = 0_W\}. \end{align*} [/definition]

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The kernel records the directions invisible to a linear map. Homogeneous linear systems behave linearly exactly because adding two solutions or rescaling one solution stays invisible to the map. Thus $\ker(T)$ is not merely a set of solutions; it is a subspace of the domain $V$, so it can be studied using bases, dimension, and coordinates.

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Kernels describe what a map collapses. To understand what a map reaches, we need the corresponding construction in the codomain. Because sums and scalar multiples of attainable outputs are again attainable outputs, the image is the natural target-side subspace attached to a linear map.

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[definition: Image of a Linear Map] Let $F$ be a field, and let $T: V \to W$ be a linear map of $F$-vector spaces. The image of $T$ is \begin{align*} \operatorname{im}(T) = \{T(v) : v \in V\}. \end{align*} [/definition]

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The image measures the effective range of a linear transformation, which may be smaller than the declared codomain. Because linear maps preserve addition and scalar multiplication, attainable outputs remain attainable after taking linear combinations. Thus $\operatorname{im}(T)$ is a subspace of $W$, and later dimension counts compare what $T$ collapses, recorded by $\ker(T)$, with what $T$ reaches, recorded by $\operatorname{im}(T)$.

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