Image and Kernel of a Linear Transformation

(1) For a function f : A→ B, we call A the domain, B the co-domain, f(A) the range, and f−1(b), for any b ∈ B, the fiber (or inverse image or pre-image) of b. For a subset S of B, f−1(B) = ⋃ b∈S f −1(b). (2) The sizes of fibers can be used to characterize injectivity (each fiber has size at most one), surjectivity (each fiber is non-empty), and bijectivity (each fiber has size exactly one). (3) Composition rules: composite of injective is injective, composite of surjective is surjective, composite of bijective is bijective. (4) If g ◦ f is injective, then f must be injective. (5) If g ◦ f is surjective, then g must be surjective. (6) If g ◦ f is bijective, then f must be injective and g must be surjective. (7) Finding the fibers for a function of one variable can be interpreted geometrically (intersect graph with a horizontal line) or algebraically (solve an equation). (8) For continuous functions of one variable defined on all of R, being injective is equivalent to being increasing throughout or decreasing throughout. More in the lecture notes, sections 2.2-2.5. (9) A vector ~v is termed a linear combination of the vectors ~ v1, ~ v2, . . . , ~ vr if there exist real numbers a1, a2, . . . , ar ∈ R such that ~v = a1 ~ v1 + a2 ~ v2 + · · · + ar ~ vr. We use the term nontrivial if the coefficients are not all zero. (10) A subspace of R is a subset that contains the zero vector and is closed under addition and scalar multiplication. (11) The span of a set of vectors is defined as the set of all vectors that can be written as linear combinations of vectors in that set. The span of any set of vectors is a subspace. (12) A spanning set for a subspace is defined as a subset of the subspace whose span is the subspace. (13) Adding more vectors either preserves or increases the span. If the new vectors are in the span of the previous vectors, it preserves the span, otherwise, it increases it. (14) The kernel and image (i.e., range) of a linear transformation are respectively subspaces of the domain and co-domain. The kernel is defined as the inverse image of the zero vector. (15) The column vectors of the matrix of a linear transformation form a spanning set for the image of that linear transformation. (16) To find a spanning set for the kernel, we convert to rref, then find the solutions parametrically (with zero as the augmenting column) then determine the vectors whose linear combinations are being discussed. The parameters serve as the coefficients for the linear combination. There is a shortening of this method. (See the lecture notes, Section 4.4, for a simple example done the long way and the short way). (17) The fibers for a linear transformation are translates of the kernel. Explicitly, the inverse image of a vector is either empty or is of the form (particular vector) + (arbitrary element of the kernel). (18) The dimension of a subspace of R is defined as the minimum possible size of a spanning set for that subspace. (19) For a linear transformation T : R → R with n ×m matrix having rank r, the dimension of the kernel is m− r and the dimension of the image is r. Full row rank r = n means surjective (image is all of R) and full column rank r = m means injective (kernel is zero subspace). (20) We can define the intersection and sum of subspaces of R.