Zero Vector¶

Definition¶

The zero vector $\boldsymbol{0}$ is the additive identity of a vector space: for every $\boldsymbol{v}\in V$ $V$ ,

\boldsymbol{v}+\boldsymbol{0}=\boldsymbol{v}.

Interpretation¶

The zero vector is the vector that changes nothing when it is added: it plays the role of the number $0$ for vector addition. In $\mathbb{R}^n$ it is the list $(0,0,\ldots,0)$ , drawn as the origin or as an arrow of length zero; in a space of matrices it is the zero matrix of that size.

The bold $\boldsymbol{0}$ and the scalar $0$ are different objects; the Key Results connect them. Every subspace must contain $\boldsymbol{0}$ , so it is the first thing to check when testing a subset: a line in $\mathbb{R}^2$ that misses the origin cannot be one. A list that contains $\boldsymbol{0}$ is never linearly independent, since $1\cdot\boldsymbol{0}=\boldsymbol{0}$ is a combination with a nonzero coefficient that gives $\boldsymbol{0}$ .

Key Results¶

The zero vector is unique.
For every vector $\boldsymbol{v}$ , $0\boldsymbol{v}=\boldsymbol{0}$ $0$ , where the $0$ on the left is the zero scalar.

Examples¶

In $\mathbb{R}^n$ , the zero vector is $(0,0,\ldots,0)$ .
For $\boldsymbol{v}=(2,-3)$ in $\mathbb{R}^2$ , $\boldsymbol{v}+(0,0)=(2,-3)$ and $\boldsymbol{v}+(-\boldsymbol{v})=(2,-3)+(-2,3)=(0,0)$ $(0, 0)$ . This checks the identity property and shows that a vector plus its additive inverse gives the zero vector.

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 2

Aliases: null vector Symbols: element of, real numbers

Last updated: 20 October 2026 03:02