Linear Combination¶

Definition¶

Let $(\boldsymbol{v}_1,\boldsymbol{v}_2,\ldots,\boldsymbol{v}_k)$ be a collection of vectors and let $\alpha_1,\alpha_2,\ldots,\alpha_k$ be scalars. A linear combination is a vector of the form

\alpha_1\boldsymbol{v}_1+\alpha_2\boldsymbol{v}_2+\cdots+\alpha_k\boldsymbol{v}_k.

Interpretation¶

A linear combination is built by scaling vectors and adding the results: scalar multiplication forms each $\alpha_i\boldsymbol{v}_i$ , and vector addition combines these scaled vectors into a single vector of the same space. These are the only two operations a vector space provides, so the linear combinations are exactly the vectors that can be made from $\boldsymbol{v}_1,\ldots,\boldsymbol{v}_k$ .

Vectors v1 and v2 scaled to 2 v1 and 1.5 v2 and added tip to tail, giving the combination 2 v1 + 1.5 v2.

The figure builds $2\boldsymbol{v}_1+1.5\boldsymbol{v}_2$ in two stages. First $\boldsymbol{v}_1$ is doubled and $\boldsymbol{v}_2$ is stretched by $1.5$ ; then the two scaled vectors are added tip to tail, so the result is the diagonal of the parallelogram they form. Other coefficients give other points. Since $\boldsymbol{v}_1$ and $\boldsymbol{v}_2$ do not lie on one line, every point of the plane is reached by some choice of $\alpha_1$ and $\alpha_2$ .

Choose $\alpha_1$ and $\alpha_2$ : the combination $\alpha_1\boldsymbol{v}_1+\alpha_2\boldsymbol{v}_2$ is reached by walking $\alpha_1\boldsymbol{v}_1$ and then $\alpha_2\boldsymbol{v}_2$ . Drag $\boldsymbol{v}_1$ and $\boldsymbol{v}_2$ to change the grid.

The coefficients may be any scalars, including $0$ and negative numbers: $\boldsymbol{v}_1-\boldsymbol{v}_2$ and $0\boldsymbol{v}_1+0\boldsymbol{v}_2=\boldsymbol{0}$ are linear combinations of $\boldsymbol{v}_1$ and $\boldsymbol{v}_2$ as well. In $\mathbb{R}^n$ , deciding whether a vector is a linear combination of given vectors means finding coefficients that satisfy one linear equation per coordinate. For example, $(3,5)=\alpha_1(1,1)+\alpha_2(1,2)$ means $\alpha_1+\alpha_2=3$ and $\alpha_1+2\alpha_2=5$ $5$ , so $\alpha_2=2$ and $\alpha_1=1$ $1$ .

Linear combinations are the common language of the next definitions: the span is the set of all of them, linear independence asks whether $\boldsymbol{0}$ is reached only with all coefficients zero, and a basis reaches every vector in exactly one way.

Key Results¶

Linear combinations are the building blocks of span, linear dependence, and bases.
Many questions about the concept reduce to solving equations involving linear combinations.

Examples¶

$2\boldsymbol{v}_1-3\boldsymbol{v}_2$ is a linear combination of $\boldsymbol{v}_1$ and $\boldsymbol{v}_2$ .

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 2
Math V: Advanced Linear Algebra · Week 5 (revisited)

Keywords: linear superposition Symbols: alpha, real numbers

Last updated: 20 October 2026 03:02