Sum of Subspaces¶

Definition¶

If $V_1,V_2,\ldots,V_m$ are subspaces of a vector space $V$, their sum is

Interpretation¶

The sum of subspaces is the set of all vectors obtainable by adding one vector from each subspace. It is the natural way to combine subspaces: the union $V_1\cup V_2$ is usually not a subspace, and the sum also includes all vectors $\boldsymbol{v}_1+\boldsymbol{v}_2$ built from the two parts.

The figure shows two planes $V_1$ and $V_2$ through the origin of $\mathbb{R}^3$ that cross in a line, their intersection $V_1\cap V_2$$V_2$. Adding a vector of $V_1$ to a vector of $V_2$ reaches every point of $\mathbb{R}^3$, so $V_1+V_2=\mathbb{R}^3$${\mathbb{R}}^3$. The dimensions confirm this: for finite-dimensional subspaces, $\dim(V_1+V_2)=\dim V_1+\dim V_2-\dim(V_1\cap V_2)$$V_2)$, here $2+2-1=3$$3$, where the line is subtracted because it is counted in both planes.

The sum is a subspace, and it is the smallest subspace containing every $V_i$. Taking $\boldsymbol{v}_2=\boldsymbol{0}$ shows $V_1\subseteq V_1+V_2$$V_2$, and any subspace containing $V_1$ and $V_2$ must contain all sums $\boldsymbol{v}_1+\boldsymbol{v}_2$${\mathbf{v}}_2$. If each $V_i$ is the span of a list of vectors, then the sum is the span of all these vectors together.

Decompositions in a sum need not be unique. In the figure, a vector on the common line can be put entirely in $V_1$ or entirely in $V_2$. When every vector of the sum has only one decomposition, the sum is called a direct sum.

Key Results¶

• The sum of subspaces is itself a subspace.
• It is the smallest subspace containing all the listed subspaces.

Examples¶

• The $x$-axis plus the $y$-axis in $\mathbb{R}^2$ equals all of $\mathbb{R}^2$.

Prerequisites¶

• Subspace

Related Concepts¶

• Direct Sum
• Linear Span
• Dimension

Sources¶

• Math II: Linear Algebra · Week 2

Keywords: Minkowski sum of subspaces Symbols: element of, intersection, real numbers, subset, union

Last updated: 22 October 2026 19:04