Linear Forms
One of the simplest types of linear transformations are those that send vectors to scalars.
Definition 7.1.1. Suppose V is a vector space over a field F. Then a linear transformation f : V β π½ is called a linear form.
Example 7.1.2. Show that the trace tr: Mn (π½) β π½ is a linear form.
Solution: We already showed in Example 2.1.2 that the trace is a linear transformation. Since it outputs scalars, it is necessarily a linear form. β
Note that the linear transformation T : β2 β β2 defined by T (x, y) = (x + 4y, 2x β 5y) for all (x, y) β β2, is not a linear form, as it does not output a scalar.
Example 7.1.3. Let C[a, b] be the vector space of continuous real-valued functions on the interval [a, b]. Show that the function I : C[a, b] β β defined by
I(f) = β«ba f(x) dx
is a linear form.
Solution. Recall that every continuous function is integrable, so this linear form makes sense. We just need to show that I is a linear transformation, since it is clear that its output is always a scalar. We thus check the two properties of Definition 7.1.1:
By properties of integrals that are typically covered in calculus courses, we know that for all f, g β C[a, b] we have
I(f + g) = β«ba (f+g)(x) dx = β«ba f(x) dx + β«ba g(x) dx = I(f)+I(g)
We similarly know that we can pull scalars in and out of integrals:
I(cf) = β«ba (cf)(x) dx = cβ«ba f(x) dx = cI(f)
∀f ∈ C[a,b] and c ∈ β. β
Theorem 7.1.4. Let B be a basis of a finite-dimensional vector space V over a field F, and let f : V β F be a linear form. Then there exists a unique vector v β V such that
f (w) = [v]TB [w]B for all w β V,
where we are treating [v]B and [w]B as column vectors.
Proof. Since f is a linear transformation, Theorem 3.3.1 tells us that it has a standard matrixβa matrix A such that f (w) = A[w]B for all w β V. Since f maps into F, which is 1-dimensional, the standard matrix A is 1 Γ n, where n = dim(V). It follows that A is a row vector, and since every vector in Fn is the coordinate vector of some vector in V, we can find some v β V such that A = [v]TB, so that f(w) = [v]TB [w]B.
Uniqueness of v follows immediately from uniqueness of standard matrices and of coordinate vectors. β‘In the special case when F = β or F = β, it makes sense to talk about the dot product of the coordinate vectors [v]B and [w]B, and the above theorem can be rephrased as saying that there exists a unique vector v β V such that f(w) = [v]B Β· [w]B for all w β V.
In the case F = β then [v]B Β· [w]B = [v]*B Β· [w]B, so we have to absorb a complex conjugate into the vector v to make this reformulation work.The space of all linear forsm on V is called "algebraic dual" of V (denoted V*).
«Epoxides Index Dual Space»