Zero Vectors, Additive Inverses, and Cancellation

The Zero Vector Is Specified by Its Coordinates

When vectors are added, their coordinates are added in corresponding positions. This coordinate-by-coordinate rule makes it possible to identify a special vector that leaves every vector unchanged under addition. It also lets us describe the vector that brings a given vector back to zero, and to determine when a common addend can be cancelled from an equation. These ideas rely on ordinary arithmetic in \(\mathbb{R}\), applied separately to each coordinate.

Throughout, \(n\in\mathbb{N}\), and vectors are columns in \(\mathbb{R}^n\). For \(\mathbf{u}=(u_1,\dots,u_n)^{\top}\) and \(\mathbf{v}=(v_1,\dots,v_n)^{\top}\), vector addition is defined by adding corresponding coordinates: \(\mathbf{u}+\mathbf{v}=(u_1+v_1,\dots,u_n+v_n)^{\top}\). Equality of vectors means equality of every corresponding coordinate. We distinguish the scalar \(0\in\mathbb{R}\) from the zero vector \(\mathbf{0}\in\mathbb{R}^n\).

The zero vector is defined using \(n\) scalar zeros, one in each coordinate. It is not the same object as a scalar zero: \(0\) is a real number, whereas \(\mathbf{0}\) is a vector with \(n\) coordinates. The coordinate formula verifies that \(\mathbf{0}\) is an additive identity. For any \(\mathbf{u}=(u_1,\dots,u_n)^{\top}\),

In the other order, \(\mathbf{0}+\mathbf{u}=(0+u_1,\dots,0+u_n)^{\top}=\mathbf{u}\), since \(0+u_i=u_i\) in \(\mathbb{R}\) for each coordinate \(i\). The zero vector works in every dimension, but the number of its coordinates must match the vectors being added.

Why the Additive Identity Is Unique

It is useful to know not only that the all-zero vector works, but that no other vector can have the same identity property. The proof uses a particular vector as input to the identity property: use the zero vector itself. This short argument illustrates how definitions can be applied directly.

Proof. The coordinate calculation above shows that \(\mathbf{0}\) is an additive identity. Suppose \(\mathbf{z}\in\mathbb{R}^n\) is any additive identity. Since \(\mathbf{z}\) is an identity, \(\mathbf{z}+\mathbf{0}=\mathbf{0}\). Since \(\mathbf{0}\) is also an identity, \(\mathbf{z}+\mathbf{0}=\mathbf{z}\). Thus \(\mathbf{z}=\mathbf{0}\), proving that every additive identity must be the zero vector. \(\square

The argument is about uniqueness, not simply the existence of a vector with zero coordinates. It rules out every possible alternative identity by showing that an identity must agree with \(\mathbf{0}\). In the same way, the additive inverse of a vector will be described by the equation that it must satisfy, and its coordinates will determine it exactly.

Constructing and Identifying an Additive Inverse

For a given vector, we seek another vector whose sum with it is the zero vector. The equation can be considered one coordinate at a time. If the \(i\)th coordinate of the original vector is \(a_i\), then the \(i\)th coordinate of its inverse must add to \(a_i\) to give zero. In real-number arithmetic, that coordinate is \(-a_i\).

To verify existence, add the proposed vector to \(\mathbf{a}\): each coordinate sum is \(a_i+(-a_i)=0\), so their vector sum is \(\mathbf{0}\). This reasoning uses the additive inverse of each real number, a fact from real-number arithmetic. It does not require a new operation on vectors: the proposed inverse is itself a vector in \(\mathbb{R}^n\).

Proof. Write \(\mathbf{a}=(a_1,\dots,a_n)^{\top}\) and form \(\mathbf{b}=(-a_1,\dots,-a_n)^{\top}\). For each \(i\), \(a_i+(-a_i)=0\), so \(\mathbf{a}+\mathbf{b}=\mathbf{0}\). Thus an additive inverse exists. To prove uniqueness, suppose \(\mathbf{c}=(c_1,\dots,c_n)^{\top}\) is another additive inverse of \(\mathbf{a}\). Then \(a_i+c_i=0\) for every \(i\). The real number \(-a_i\) is the unique real number that adds to \(a_i\) to give \(0\), so \(c_i=-a_i\) for every \(i\). Hence \(\mathbf{c}=\mathbf{b}\), proving uniqueness. \(\square

Writing the inverse in coordinates is important: it gives both a construction and a test. To check a proposed inverse, add coordinate by coordinate and confirm that every result is zero. A single nonzero coordinate in the sum means the proposed vector is not an inverse.

Cancellation Follows Coordinate by Coordinate

Cancellation in vector addition is justified when the same vector is added to both sides of an equation. The reason is not a special manipulation of vectors: equality of vectors gives equality in each coordinate, and real-number addition allows a common real addend to be cancelled in each such equality. The matching of the addends is essential.

Proof. For the first implication, write the vectors using coordinates \(u_i,v_i,w_i\). Equality of the two sums means that \(u_i+w_i=v_i+w_i\) for every \(i\in\{1,\dots,n\}\). Cancellation of the common real number \(w_i\) gives \(u_i=v_i\) for every \(i\). Therefore the vectors have equal corresponding coordinates, so \(\mathbf{u}=\mathbf{v}\). For the second implication, equality \(\mathbf{w}+\mathbf{u}=\mathbf{w}+\mathbf{v}\) gives \(w_i+u_i=w_i+v_i\) for every \(i\). Real-number cancellation gives \(u_i=v_i\) for every \(i\), and again coordinate equality yields \(\mathbf{u}=\mathbf{v}\). \(\square

This proof states both directions of the cancellation claim: a common addend on the right can be cancelled, and a common addend on the left can be cancelled. It does not assume any vector cancellation rule in order to prove one. Each direction follows from the definition of vector addition together with the corresponding real-number cancellation property.

Common Pitfall: The Addends Must Match

A common error is to cancel two addends merely because both sides of an equation are sums. Cancellation requires the same vector to occur as an addend on both sides. If the addends differ, the cancellation theorem does not apply, even if the two vector sums happen to be equal.

For example, the following equation is valid:

A wrong calculation would “cancel” \(\begin{bmatrix}2\\0\end{bmatrix}\) on the left against \(\begin{bmatrix}3\\0\end{bmatrix}\) on the right and conclude that \(\begin{bmatrix}1\\0\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}\). That conclusion is false: the first coordinates are \(1\) and \(0\). The correction is to check the hypothesis before cancelling. The common addends here are not equal, so the cancellation theorem gives no conclusion about the remaining vectors.

There is a related notation pitfall: do not write \(0\) when the equation calls for the zero vector. In \(\mathbb{R}^2\), for example, an addition such as \(\mathbf{u}+\mathbf{0}\) uses \(\mathbf{0}=\begin{bmatrix}0\\0\end{bmatrix}\), not a scalar by itself. Keeping the dimension and the bold vector notation visible helps ensure that each sum is well-defined.

The coordinate arguments in this tutorial provide a basic method for checking claims about vector addition: state the vector equation, translate it into an equation in each coordinate, and use the relevant fact about real numbers. Here the method establishes the identity vector, constructs and identifies inverses, and proves exactly when cancellation is valid.