Two vectors of the same length are added one position at a time, and a vector is scaled by multiplying every component by a single number. Both results stay in the same space.

Two shopping lists with the same items add up column by column. Cooking for twice as many guests multiplies every quantity by the same two.

Addition is defined only between vectors with the same number of components, and it acts separately in each position. A scalar is an ordinary real number; multiplying a vector by it stretches every component by the same factor. Both operations produce a vector in the same $\mathbb{R}^n$ they started in.

• $\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}$, because each position adds two real numbers and that addition is commutative.
• $\mathbf{v}+\mathbf{0}=\mathbf{v}$, where $\mathbf{0}$ is the zero vector with as many components as $\mathbf{v}$.
• $(-1)\mathbf{v}$ is written $-\mathbf{v}$, and $\mathbf{v}+(-\mathbf{v})=\mathbf{0}$.
• $0\mathbf{v}=\mathbf{0}$: a scalar zero on the left, a vector of zeros on the right.
• $(c+d)\mathbf{v}=c\mathbf{v}+d\mathbf{v}$, again because each position is an instance of the rule for real numbers.

The large triangle is the small one at twice the size. Doubling $\mathbf{u}$ and $\mathbf{v}$ and then adding reaches the same place as adding first and doubling after, which is what $c(\mathbf{u}+\mathbf{v})=c\mathbf{u}+c\mathbf{v}$ says.

Both sides are vectors with the same number of components, so by the definition of equality they are equal exactly when they agree in every position. In position i the left side is $c(u_i+v_i)$ and the right side is $cu_i+cv_i$, and those two real numbers are equal by the distributive law for real numbers.

Fix two vectors with the same number of components and one scalar.

Read position i of the left-hand side, using first the definition of addition and then the definition of scaling.

These are three real numbers, so the distributive law for real numbers applies to them.

The same number is position i of the right-hand side, by the two definitions read in the other order.

Both sides have n components and agree in every position, which is what equality of vectors demands.

Position 1 gives $1+4$, position 2 gives $7+(-2)$.

The middle component is $5\times 0=0$, which is the scaling rule applied to zero like any other number.

Position 3 exists on one side only, so this expression names nothing.

A number goes in on the left and a vector comes out on the right.