Learn Vector Spaces & Subspaces: Interactive Problems and Worked Examples

Vector spaces, subspaces, and the subspace test

Key idea

A vector space \(V\) over a field (like \(\mathbb{R}\)) is a set with two operations: vector addition and scalar multiplication, satisfying the standard axioms (associativity, commutativity of addition, distributive laws, scalar identity, additive identity \(0\), additive inverses, and closure).

A subset \(U\subseteq V\) is a subspace if it is a vector space using the same operations. In practice, use the subspace test.

The subspace test

• Zero vector: \(0 \in U\).
• Closed under addition: if \(u,v\in U\), then \(u+v\in U\).
• Closed under scalar multiplication: if \(u\in U\) and \(c\in \mathbb{R}\), then \(cu\in U\).

Common examples

• \(\mathbb{R}^n\) and any plane through the origin in \(\mathbb{R}^3\).
• Matrix spaces like \(M_{m\times n}(\mathbb{R})\); sets defined by linear constraints are often subspaces.
• Function spaces like \(C[0,1]\); sets defined by linear conditions (e.g. \(f(0)=0\)) are subspaces.

Worked example

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Summary

• A subspace must contain \(0\) and be closed under addition and scalar multiplication.
• Sets like \(x+y=1\) are usually affine (shifted) and fail the \(0\in U\) test.

Linear combinations, span, and describing subspaces

Key idea

A linear combination of vectors \(v_1,\dots,v_k\) is any vector of the form \[ c_1 v_1 + \cdots + c_k v_k \] where \(c_1,\dots,c_k\in\mathbb{R}\). The span is the set of all such combinations: \[ \text{span}\{v_1,\dots,v_k\}=\{c_1 v_1 + \cdots + c_k v_k : c_i\in\mathbb{R}\}. \] A span is always a subspace.

Solution spaces are subspaces

The solution set of a homogeneous system \(Ax=0\) is the null space of \(A\), \[ \mathcal{N}(A)=\{x: Ax=0\}, \] and it is always a subspace of \(\mathbb{R}^n\).

Worked example

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Summary

• \(\text{span}(S)\) is always a subspace: it contains \(0\) and is closed under linear combinations.
• Homogeneous solution spaces (null spaces) are subspaces and their dimension equals the number of free parameters.

Bases, coordinates relative to a basis, and dimension

Key idea

A set \(B=\{b_1,\dots,b_n\}\) is a basis for a vector space \(V\) if: (1) it spans \(V\), and (2) it is linearly independent. When \(B\) is a basis, every \(v\in V\) can be written uniquely as \[ v = c_1 b_1 + \cdots + c_n b_n. \] The scalars \((c_1,\dots,c_n)\) are the coordinates of \(v\) relative to the basis \(B\).

Dimension

If \(V\) is finite-dimensional, the dimension \(\dim V\) is the number of vectors in any basis of \(V\). For example, \(\dim \mathbb{R}^n = n\), \(\dim P_n = n+1\), and \(\dim M_{m\times n}(\mathbb{R}) = mn\). If \(V\) has a basis of size \(5\), then \(\dim V = 5\) and \(\dim V^* = 5\) as well.

Worked example

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Summary

• Bases give unique coordinate representations.
• Dimension equals the number of vectors in a basis (and \(\dim V^*=\dim V\) for finite-dimensional \(V\)).

Intersection, sum, union, and common subspace geometry in \(\mathbb{R}^n\)

Intersection \(U\cap W\)

If \(U\) and \(W\) are subspaces of \(V\), then \(U\cap W\) is always a subspace of \(V\). Reason: it contains \(0\) and is closed under addition/scalar multiplication because both \(U\) and \(W\) are.

Sum \(U+W\)

The sum of subspaces is \[ U+W=\{u+w: u\in U,\; w\in W\}. \] It is the smallest subspace containing both \(U\) and \(W\).

Union \(U\cup W\)

In general, the union of two subspaces is not a subspace. It is a subspace only in the special case where one subspace is contained in the other: \[ U\cup W \text{ is a subspace } \Longleftrightarrow U\subseteq W \text{ or } W\subseteq U. \]

Worked example

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Summary

• \(U\cap W\) and \(U+W\) are always subspaces.
• \(U\cup W\) is usually not a subspace (unless one contains the other).

Dimension in practice: constraints, degrees of freedom, and infinite dimension

Key idea

Dimension is a “degrees of freedom” count. Every independent linear constraint typically reduces dimension by \(1\) (in finite-dimensional settings). For example, a single homogeneous linear equation in \(\mathbb{R}^3\) defines a 2D subspace (a plane through the origin).

Finite vs. infinite dimension

A vector space is finite-dimensional if it has a finite basis. If no finite basis exists, it is infinite-dimensional (for example, \(P\) = all polynomials, or \(C[0,1]\)). If a subspace \(S\) has infinite dimension, then it cannot be spanned by finitely many vectors (every spanning set must be infinite).

Worked example

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Summary

• Dimensions count free parameters (degrees of freedom).
• Infinite-dimensional means no finite spanning set exists.

Quotient spaces \(V/W\): vectors modulo a subspace

Key idea

Let \(W\) be a subspace of \(V\). Two vectors \(v\) and \(u\) are considered equivalent modulo \(W\) if \[ v-u \in W. \] The equivalence class of \(v\) is the coset \[ v+W = \{v+w : w\in W\}. \] The quotient space \(V/W\) is the set of all cosets: \[ V/W = \{v+W : v\in V\}. \]

How to think about \(V/W\)

• You “collapse” the entire subspace \(W\) to act like the zero element in the quotient.
• Vectors that differ by an element of \(W\) become the same point in \(V/W\).
• This is useful for focusing on directions “not in \(W\)” and for simplifying structure.

Worked example

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Summary

• \(V/W\) is the set of cosets \(v+W\), i.e. vectors modulo the subspace \(W\).
• Quotients “collapse” directions in \(W\) so you focus on what remains.

Why vector spaces and subspaces matter

Where vector spaces and subspaces show up

• Linear systems: solution sets of \(Ax=0\) are subspaces (null spaces).
• Linear transformations: kernels and images are subspaces; dimension relates to rank and nullity.
• Geometry: lines/planes through the origin are subspaces; sums and intersections match geometric intuition.
• Data and ML: subspaces model low-dimensional structure inside high-dimensional data (PCA).
• Functions: many spaces of functions are vector spaces; constraints like \(f(0)=0\) define subspaces.

Worked example: a clean dimension count in matrices

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Final recap

• Subspace test: check \(0\in U\), closure under addition, closure under scalar multiplication.
• Span: all linear combinations; \(\text{span}(S)\) is always a subspace.
• Basis: spans + independent; gives unique coordinates.
• Dimension: number of basis vectors; counts degrees of freedom.
• Operations: \(U\cap W\) and \(U+W\) are subspaces; \(U\cup W\) usually is not.
• Quotient: \(V/W\) is cosets \(v+W\), i.e. vectors modulo \(W\).