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†’ 𝔽 is called a linear form.

Example 7.1.2. Show that the trace tr: Mn (𝔽) β†’ 𝔽 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 : ℝ2 β†’ ℝ2 defined by T (x, y) = (x + 4y, 2x βˆ’ 5y) for all (x, y) ∈ ℝ2, 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†’ ℝ defined by

I(f) = ∫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:

  1. By properties of integrals that are typically covered in calculus courses, we know that for all f, g ∈ C[a, b] we have

    I(f + g) = ∫ba (f+g)(x) dx = ∫ba f(x) dx + ∫ba g(x) dx = I(f)+I(g)

  2. We similarly know that we can pull scalars in and out of integrals:

    I(cf) = ∫ba (cf)(x) dx = c∫ba f(x) dx = cI(f)

    ∀f ∈ C[a,b] and c ∈ ℝ.  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 ∈ V such that

    f (w) = [v]TB [w]B for all w ∈ 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 matrixβ€”a matrix A such that f (w) = A[w]B for all w ∈ 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 ∈ 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 = ℝ 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 ∈ V such that f(w) = [v]B Β· [w]B for all w ∈ 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»