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A scalar multiplying a specified term in a linear combination, polynomial or power series. When the term family has linear independence, the coefficients are unique; uniqueness of finite Laurent polynomial coefficients follows after multiplying by a sufficiently large power to obtain an ordinary polynomial.

A convex cone is a subset $C$ of a real vector space closed under nonnegative linear combinations: $ax+by\in C$ whenever $x,y\in C$ and $a,b\ge 0$. The positive semidefinite cone and the copositive cone are examples, ordered by inclusion through the positive-semidefinite-plus-nonnegative cone.

A Gaussian random vector has a multivariate normal distribution; equivalently, every real linear combination of its coordinates is a normal random variable, allowing zero variance. Its expected value vector and covariance matrix determine its law. Subvectors are independent when their cross-covariances vanish, a feature particular to joint Gaussianity.

A Jordan chain of a linear operator $T$ for an eigenvalue $\lambda$ is a list of nonzero vectors satisfying $(T-\lambda I)v_1=0$ and $(T-\lambda I)v_j=v_{j-1}$ for $2\le j\le r$. These vectors are linearly independent: in a vanishing linear combination, applying the largest relevant power of $T-\lambda I$ isolates the last coefficient times $v_1$, then descending induction removes all coefficients. On their span the operator has one Jordan block. This is different from a generalized eigenfunction that solves $(T-\lambda I)u=0$ outside the original function space.

The linear span of a subset $S$ of a vector space is the set of all finite linear combinations of elements of $S$.

A Gaussian velocity profile at each position, with coefficients encoding local density, mean velocity and temperature. Its logarithm is a linear combination of collision invariants, so its collision term vanishes pointwise in position. Transport need not vanish: the coefficients must also satisfy their transport equations for a local Maxwellian to solve the full Boltzmann equation.

When $R^2>4L/C$, the homogeneous series RLC circuit has two distinct negative characteristic roots, $[-R\pm \sqrt{R^2-4L/C}]/(2L)$. Its current is a linear combination of the corresponding decaying exponential functions.