Vectors in Rⁿ and Linear Combinations

The column picture turned a system into a question about combining vectors. This topic sets up the language for that, in any number of dimensions.

A vector in $\mathbb{R}^n$ is an ordered list of $n$ real numbers, written as a column: $\mathbf{v} = \begin{bmatrix} v_1 \\ \vdots \\ v_n \end{bmatrix}$. Two vectors are equal when every entry matches. We add vectors entry by entry and multiply a vector by a number (a scalar) by multiplying every entry. For $n = 2$ or $3$ you can draw them as arrows from the origin; for larger $n$ the algebra works the same even though the picture runs out.

In the plane, adding $\mathbf{u} + \mathbf{v}$ means walking along $\mathbf{u}$ and then along $\mathbf{v}$ (tip to tail); the sum is the diagonal of the parallelogram the two arrows make. Scaling by $c$ stretches an arrow by $|c|$ and flips it when $c < 0$. The usual rules hold because they hold in each entry: $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$, $c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v}$, $(c + d)\mathbf{u} = c\mathbf{u} + d\mathbf{u}$, and $\mathbf{u} + \mathbf{0} = \mathbf{u}$.

Given vectors $\mathbf{v}_1, \dots, \mathbf{v}_p$ and scalars $c_1, \dots, c_p$, the vector $$c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_p\mathbf{v}_p$$ is a linear combination of the $\mathbf{v}$'s with weights $c_1, \dots, c_p$. Weights can be any real numbers, including $0$ and negatives.