Proving the Algebraic Laws of Rn

Algebraic Laws from Coordinates

Calculations with vectors often use familiar algebraic steps: changing the order of two vectors being added, changing the grouping of a sum, or moving a scalar inside a sum. These steps are not separate rules invented for vectors. They follow from the definitions of vector addition and scalar multiplication, together with the corresponding laws for real numbers. A coordinatewise proof makes that connection precise.

Fix a positive integer \(n\), and let \(\mathbf{u},\mathbf{v},\mathbf{w}\in\mathbb{R}^n\) and \(a,b\in\mathbb{R}\). Write the vectors as columns: \(\mathbf{u}=(u_1,\dots,u_n)^{\top}\), \(\mathbf{v}=(v_1,\dots,v_n)^{\top}\), and \(\mathbf{w}=(w_1,\dots,w_n)^{\top}\). Their coordinates are real numbers. Addition and scalar multiplication are defined coordinate by coordinate:

Two vectors in \(\mathbb{R}^n\) are equal exactly when their corresponding coordinates are equal. Thus, to prove an equality of vectors, it is enough to take an arbitrary coordinate index \(i\), where \(1\le i\le n\), and prove that the \(i\)th coordinate of the left side equals the \(i\)th coordinate of the right side. The real-number laws used in each coordinate are commutativity and associativity of addition, distributivity of multiplication over addition, and associativity of multiplication.

The Four Laws and Their Proof

Proof. Let \(i\) be any coordinate index with \(1\le i\le n\). By the definition of vector addition, the \(i\)th coordinate of \(\mathbf{u}+\mathbf{v}\) is \(u_i+v_i\), while the \(i\)th coordinate of \(\mathbf{v}+\mathbf{u}\) is \(v_i+u_i\). Real addition is commutative, so \(u_i+v_i=v_i+u_i\). This holds for every \(i\), hence \(\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}\).

For associativity of vector addition, the \(i\)th coordinate of \((\mathbf{u}+\mathbf{v})+\mathbf{w}\) is \((u_i+v_i)+w_i\). The \(i\)th coordinate of \(\mathbf{u}+(\mathbf{v}+\mathbf{w})\) is \(u_i+(v_i+w_i)\). Real addition is associative, so these coordinates are equal for every \(i\). Therefore \((\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})\).

For distributivity of scalar multiplication over vector addition, the \(i\)th coordinate of \(a(\mathbf{u}+\mathbf{v})\) is \(a(u_i+v_i)\). The \(i\)th coordinate of \(a\mathbf{u}+a\mathbf{v}\) is \(au_i+av_i\). Distributivity in \(\mathbb{R}\) gives \(a(u_i+v_i)=au_i+av_i\) for every \(i\), so \(a(\mathbf{u}+\mathbf{v})=a\mathbf{u}+a\mathbf{v}\).

Finally, the \(i\)th coordinate of \(a(b\mathbf{u})\) is \(a(bu_i)\), and the \(i\)th coordinate of \((ab)\mathbf{u}\) is \((ab)u_i\). Associativity of real multiplication gives \(a(bu_i)=(ab)u_i\) for every \(i\). Thus \(a(b\mathbf{u})=(ab)\mathbf{u}\). Since each claimed equality holds in every coordinate, all four vector equalities follow. \(\square\)

The fourth law is sometimes called the associativity of scalar multiplication. It does not say that vectors are being multiplied by one another: \(a\) and \(b\) are real scalars, and \(ab\) is their ordinary real product. The parentheses indicate two equivalent ways to apply the scalar multiplications.

Checking the Laws in Examples

Two Useful Consequences

The laws also explain what happens when the scalar or vector being scaled is zero. Here \(\mathbf{0}\) denotes the zero vector in \(\mathbb{R}^n\), whose every coordinate is \(0\); the scalar \(0\) is the real number zero. These are different objects, even though their coordinates are connected by the definition of scalar multiplication.

Proof. For each coordinate \(i\), the \(i\)th coordinate of \(0\mathbf{u}\) is \(0u_i=0\), which is the \(i\)th coordinate of \(\mathbf{0}\). Therefore \(0\mathbf{u}=\mathbf{0}\). The \(i\)th coordinate of \(a\mathbf{0}\) is \(a\cdot0=0\), again the \(i\)th coordinate of \(\mathbf{0}\). Therefore \(a\mathbf{0}=\mathbf{0}\). \(\square\)

Another consequence is that multiplying a vector by the product of two scalars can be evaluated in either order. The theorem proves this for arbitrary real \(a\) and \(b\), not merely for positive integers. Consequently, grouping is safe whenever the operations have the same meaning and the expressions are defined. It does not authorize changing the order of unrelated operations or replacing a vector by a scalar.

Common Mistake: Distributing to Only One Term

A frequent error is to multiply just one term in a vector sum by the scalar. For example, with \(\mathbf{u}=(1,3)^{\top}\), \(\mathbf{v}=(4,2)^{\top}\), and scalar \(2\), the incorrect calculation is \(2(\mathbf{u}+\mathbf{v})=2\mathbf{u}+\mathbf{v}\). It produces

But the left side of the claimed equality is actually

The vectors \((6,8)^{\top}\) and \((10,10)^{\top}\) are not equal. The correction is to distribute the scalar to both vectors: \(2(\mathbf{u}+\mathbf{v})=2\mathbf{u}+2\mathbf{v}\). Indeed, \(2\mathbf{u}=(2,6)^{\top}\) and \(2\mathbf{v}=(8,4)^{\top}\), whose sum is \((10,10)^{\top}\).

Why Coordinatewise Proofs Matter

Coordinatewise reasoning turns a vector identity into a collection of real-number identities. This is why ordinary algebraic rearrangements can be used in vector calculations: the vector laws hold because the corresponding real laws hold in every coordinate. The same method also provides a reliable way to check a proposed identity. If even one coordinate differs, the vectors are not equal; if every coordinate agrees, the vectors are equal by the definition of equality in \(\mathbb{R}^n\).

When writing a proof, keep the objects and operations distinct. Vector addition combines corresponding coordinates. Scalar multiplication multiplies every coordinate by one real number. The scalar product \(ab\) is multiplication in \(\mathbb{R}\), while \(a\mathbf{u}\) is scalar multiplication of a vector. Naming these steps and showing the \(i\)th coordinate prevents a familiar-looking expression from hiding an unjustified change.