Subspace Intersection and Union Calculator

Math Linear Algebra • Vector Spaces and Linear Independence

Written by STEM Calculators Team Published February 22, 2026 Updated February 24, 2026

Compute bases for the intersection \(V\cap W\) and the sum \(V+W=\{v+w:\,v\in V,w\in W\}\), where \(V=\mathrm{span}\{v_1,\dots,v_p\}\) and \(W=\mathrm{span}\{w_1,\dots,w_q\}\) in \(\mathbb{R}^d\). The intersection is found by solving \(Va=Wb\) via the nullspace of \([V\;|\;-W]\).

Dimension \(d\)

# generators for \(V\) (p)

# generators for \(W\) (q)

Input orientation

Tolerance

Display decimals

Steps

Show row-ops log (RREF) Show nullspace basis for \([V\;|\;-W]\) Simplify coefficients to integers (when possible) Show 2D plot when \(d=2\)

Inputs accept -3.5, 2e-4, fractions like 7/3, and constants pi, e. Note: the set union \(V\cup W\) is generally not a subspace, so this tool computes the sum \(V+W\).

Generators for \(V\)

V matrix is 2×1

Columns are \(v_1,\dots,v_p\).

Generators for \(W\)

W matrix is 2×1

Columns are \(w_1,\dots,w_q\).

Results

\(\dim(V)\)

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\(\dim(W)\)

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\(\dim(V\cap W)\)

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\(\dim(V+W)\)

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Reduced basis for \(V\)

Pivot generators from \(V\) form a basis for \(V\).

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Reduced basis for \(W\)

Pivot generators from \(W\) form a basis for \(W\).

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Basis for \(V+W\)

Pivot columns from \([V\;W]\) give a basis for the sum.

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Basis for \(V\cap W\)

Computed from nullspace vectors \(z=(a,b)\) of \([V\;|\;-W]\), then \(x=Va=Wb\).

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Ready

RREF panels

Shows pivots for \(V\), \(W\), \([V\;W]\), and \([V\;|\;-W]\).

RREF(V)

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RREF(W)

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RREF([V W])

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RREF([V | -W])

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Nullspace basis for \([V\;|\;-W]\)

Each basis vector \(z=(a,b)\) produces an intersection vector \(x=Va=Wb\).

Enter generators and click “Calculate”.

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Theory

Subspace sum and intersection

Let \(V\) and \(W\) be subspaces of \(\mathbb{R}^d\). Their intersection is \[ V\cap W=\{x:\ x\in V\ \text{and}\ x\in W\}, \] which is always a subspace.

Their sum is \[ V+W=\{v+w:\ v\in V,\ w\in W\}, \] also always a subspace. (By contrast, the set union \(V\cup W\) is usually not a subspace unless one subspace contains the other.)

Computing \(V+W\) from generators

If \(V=\mathrm{span}\{v_1,\dots,v_p\}\) and \(W=\mathrm{span}\{w_1,\dots,w_q\}\), then \[ V+W=\mathrm{span}\{v_1,\dots,v_p,w_1,\dots,w_q\}. \] Put all generators into one matrix \([V\;W]\). The pivot columns of \([V\;W]\) (after RREF) correspond to a basis for \(V+W\), and \[ \dim(V+W)=\mathrm{rank}([V\;W]). \]

Computing \(V\cap W\) via a nullspace

A vector \(x\) lies in \(V\cap W\) exactly when it can be written both ways: \[ x = Va = Wb \] for some coefficient vectors \(a\in\mathbb{R}^p\) and \(b\in\mathbb{R}^q\). Rearranging gives \[ Va - Wb = 0 \quad\Longleftrightarrow\quad [V\;|\;-W]\begin{bmatrix}a\\ b\end{bmatrix}=0. \] Therefore, if \(z=(a,b)\) is a nullspace vector of \([V\;|\;-W]\), then \[ x = Va = Wb \] is an intersection vector. A basis for \(V\cap W\) is obtained by generating these \(x\)-vectors from a nullspace basis and reducing them to an independent subset.

Grassmann dimension formula

The dimensions satisfy the Grassmann formula: \[ \dim(V+W)=\dim(V)+\dim(W)-\dim(V\cap W). \] Intuitively: add the dimensions, then subtract the overlap counted twice. The calculator computes all terms and displays a numerical check.

2D intuition

In \(\mathbb{R}^2\), the only subspaces are \(\{0\}\), lines through the origin, and \(\mathbb{R}^2\) itself. So:

If \(V\) and \(W\) are different lines, then \(V\cap W=\{0\}\) and \(V+W=\mathbb{R}^2\).
If they are the same line, then \(V\cap W=V=W\) and \(V+W=V\).
If one is \(\mathbb{R}^2\), then the sum is \(\mathbb{R}^2\) and the intersection is the smaller subspace.

University note: infinite-dimensional spaces

The ideas extend to function spaces and other infinite-dimensional vector spaces. The intersection and sum are still subspaces, and the Grassmann identity has analogs under suitable finiteness assumptions. Computation, however, depends on the structure (inner products, orthogonality, projections, and operator methods).

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