Linear Combinations & Span

Imagine you have a few basic "ingredient" vectors, and you are allowed to stretch each one (multiply it by a scalar) and then add the stretched pieces together. Anything you can build this way is a linear combination.

Formally, given vectors $v_1, v_2, \dots, v_k$ in $\mathbb{R}^n$ and scalars (real numbers) $c_1, c_2, \dots, c_k$, the vector $c_1 v_1 + c_2 v_2 + \dots + c_k v_k$ is called a linear combination of $v_1, \dots, v_k$. The numbers $c_i$ are the weights (or coefficients).

• Each $v_i$ is a vector — a column of $n$ real numbers.
• Each $c_i$ is a single real number.
• Only two operations are ever used: scalar multiplication and vector addition.

For example, with $v_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $v_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$, the combination $3 v_1 + 2 v_2 = \begin{bmatrix} 3 \\ 2 \end{bmatrix}$. Choosing different weights lands you on different points.

If a single linear combination is one destination, the span is the map of every destination you could possibly reach.

The span of $v_1, \dots, v_k$, written $\operatorname{Span}\{v_1, \dots, v_k\}$, is the set of all linear combinations of those vectors: $\operatorname{Span}\{v_1, \dots, v_k\} = \{\, c_1 v_1 + \dots + c_k v_k \;:\; c_1, \dots, c_k \in \mathbb{R} \,\}.$ As the weights $c_i$ range over all real numbers, you sweep out the entire span.

• A vector $w$ is in the span exactly when you can find weights with $w = c_1 v_1 + \dots + c_k v_k$.
• So checking membership is the same as asking whether a particular system of linear equations has a solution.

Note that a span is usually an infinite set, even though it is generated by only finitely many vectors.

A subspace of $\mathbb{R}^n$ is a set $W$ that behaves well under the two operations. Concretely, $W$ is a subspace when:

1. $\mathbf{0} \in W$ (it contains the zero vector),
2. if $u, w \in W$ then $u + w \in W$ (closed under addition),
3. if $u \in W$ and $c \in \mathbb{R}$ then $c u \in W$ (closed under scalar multiplication).

Key fact: $\operatorname{Span}\{v_1, \dots, v_k\}$ automatically satisfies all three, so a span is always a subspace.

• Contains $\mathbf{0}$: choose every weight $c_i = 0$, giving $0 v_1 + \dots + 0 v_k = \mathbf{0}$.
• Closed under addition: adding two linear combinations gives another linear combination (just add the matching weights).
• Closed under scalar multiplication: scaling a linear combination by $c$ simply scales every weight by $c$.

Because every subspace must contain $\mathbf{0}$, a span can never be a line or plane that misses the origin.

The shape of a span is determined by how many independent directions its generating vectors supply.

• Only the zero vector: $\operatorname{Span}\{\mathbf{0}\} = \{\mathbf{0}\}$ — just the origin, a single point.
• One nonzero vector $v$: $\operatorname{Span}\{v\}$ is the line through the origin in the direction of $v$ (all multiples $c v$).
• Two linearly independent vectors: their span is a plane through the origin.
• $n$ linearly independent vectors in $\mathbb{R}^n$: their span is all of $\mathbb{R}^n$.

Watch for redundancy: if one vector is just a multiple of another (they are parallel), it adds no new direction, so the two of them together still span only a line. Here "independent" means each new vector points in a genuinely new direction that the others cannot produce.