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Linear Functional — Definition, Formula & Examples

A linear functional is a function that takes a vector as input and returns a single scalar, while respecting both vector addition and scalar multiplication. It maps an entire vector space down to the real (or complex) numbers.

A linear functional on a vector space VV over a field FF is a linear map f:V→Ff: V \to F. That is, for all vectors u,v∈V\mathbf{u}, \mathbf{v} \in V and all scalars c∈Fc \in F, the map satisfies f(u+v)=f(u)+f(v)f(\mathbf{u} + \mathbf{v}) = f(\mathbf{u}) + f(\mathbf{v}) and f(cu)=c f(u)f(c\mathbf{u}) = c\,f(\mathbf{u}).

Key Formula

f(x)=a1x1+a2x2+⋯+anxnf(\mathbf{x}) = a_1 x_1 + a_2 x_2 + \cdots + a_n x_n
Where:
  • ff = The linear functional mapping vectors to scalars
  • aia_i = Fixed scalar coefficients defining the functional
  • xix_i = Components of the input vector \mathbf{x}

How It Works

You can think of a linear functional as a single row vector acting on column vectors via the dot product. Given a row vector aT=[a1,a2,…,an]\mathbf{a}^T = [a_1, a_2, \ldots, a_n], the map f(x)=aTxf(\mathbf{x}) = \mathbf{a}^T \mathbf{x} is a linear functional on Rn\mathbb{R}^n. Every linear functional on a finite-dimensional space can be represented this way once you fix a basis. The collection of all linear functionals on VV forms its own vector space, called the dual space V∗V^*.

Worked Example

Problem: Let f:R3→Rf: \mathbb{R}^3 \to \mathbb{R} be defined by f(x)=2x1−3x2+x3f(\mathbf{x}) = 2x_1 - 3x_2 + x_3. Verify that ff is a linear functional by checking both properties for u=(1,0,4)\mathbf{u} = (1, 0, 4) and v=(2,1,−1)\mathbf{v} = (2, 1, -1) with scalar c=5c = 5.
Compute f(u) and f(v): Evaluate the functional on each vector separately.
f(u)=2(1)−3(0)+1(4)=6,f(v)=2(2)−3(1)+1(−1)=0f(\mathbf{u}) = 2(1) - 3(0) + 1(4) = 6, \quad f(\mathbf{v}) = 2(2) - 3(1) + 1(-1) = 0
Check additivity: Compute u+v=(3,1,3)\mathbf{u} + \mathbf{v} = (3, 1, 3) and evaluate ff.
f(u+v)=2(3)−3(1)+1(3)=6=6+0=f(u)+f(v)✓f(\mathbf{u}+\mathbf{v}) = 2(3) - 3(1) + 1(3) = 6 = 6 + 0 = f(\mathbf{u}) + f(\mathbf{v}) \checkmark
Check scalar homogeneity: Compute cu=(5,0,20)c\mathbf{u} = (5, 0, 20) and evaluate ff.
f(5u)=2(5)−3(0)+1(20)=30=5⋅6=c f(u)✓f(5\mathbf{u}) = 2(5) - 3(0) + 1(20) = 30 = 5 \cdot 6 = c\,f(\mathbf{u}) \checkmark
Answer: Both linearity conditions hold, confirming that f(x)=2x1−3x2+x3f(\mathbf{x}) = 2x_1 - 3x_2 + x_3 is a linear functional on R3\mathbb{R}^3.

Why It Matters

Linear functionals appear throughout applied mathematics and physics. In optimization, the objective function in a linear program is a linear functional. In quantum mechanics, bra vectors are linear functionals on the state space, making the concept essential for anyone studying advanced science or engineering.

Common Mistakes

Mistake: Confusing a linear functional with a general linear transformation. Students sometimes think any linear map T:V→WT: V \to W is a linear functional.
Correction: A linear functional specifically maps into the scalar field (W=FW = F), not into an arbitrary vector space. If the output is a vector rather than a scalar, it is a linear transformation but not a linear functional.