What Vol I left out
A first course studies vectors, a basis, and linear transformations between spaces. But there is one transformation it usually rushes past: the map that eats a vector and spits out a single number. A linear functional on a vector space V is a linear map f: V -> F into the scalar field. It does not move you to another vector — it measures you.
Linearity means f(au + bv) = a·f(u) + b·f(v). That is the whole rule. The familiar dot product against a fixed vector is the prototype: in R^3, f(x) = 3x1 - x2 + 2x3 is a functional. So is 'read off the second coordinate'. So is 'take the trace' on the space of matrices, and 'integrate from 0 to 1' on a space of functions.
Functionals form a space
Add two functionals pointwise, scale a functional — the result is still a functional. So the set of all linear functionals on V is itself a vector space, called the dual space and written V*. Its 'vectors' are measurements. To stress the contrast we call an element of V* a covector: vectors are things to be measured, covectors do the measuring.
The level set f = 0 of a nonzero functional is a hyperplane through the origin — a subspace one dimension short of all of V. Every functional is essentially 'how far past this hyperplane are you, measured in fixed steps'. This geometric view is what column vectors can never quite show you.
A worked functional
V = R^3, functional f(x) = 3*x1 - 1*x2 + 2*x3
As a row vector: f = [ 3 -1 2 ]
Apply to column: v = [ 4 ; 5 ; 1 ]
f(v) = [3 -1 2] * [4; 5; 1]
= 3*4 + (-1)*5 + 2*1
= 12 - 5 + 2 = 9
Hyperplane f = 0: 3x1 - x2 + 2x3 = 0 (a plane through origin)
Level f = 9 is the parallel plane that v sits on.Notice the shape: the functional is written as a row, the vector as a column, and matrix multiplication of row-by-column lands you in F. That typographic asymmetry is not an accident — it is the first hint that V and V* are different kinds of object, the theme of this whole track.