Throughout this section, V is a subspace of K^((m)), W is a subspace of
K^((n)) and a is a linear transformation belonging to Hom_(K)(V, W).
v * a : ModTupFldElt, ModMatFldElt -> ModTupFldElt
Given an element v belonging to the vector space V, and an element a belonging to Hom_(K)(V, W), return the image of v under the linear transformation a as an element of the vector space W.
Given a matrix a belonging to K^((m x n)) and a matrix b belonging to K^((n x p)), for some integers m, n, p, form the product of a and b as an element of K^((m x p)).
The domain of the linear transformation a belonging to Hom_(K)(V, W), returned as a subspace of V.
The codomain of the linear transformation a belonging to Hom_(K)(V, W), returned as a subspace of W.
The image of the linear transformation a belonging to Hom_(K)(V, W), returned as a subspace of W.
The dimension of the image of the linear transformation a, i.e. the row rank of the matrix a.
The (row) echelon form of matrix a belonging to Hom_(K)(V, W). This function returns two values:
- The (row) echelon form e of a; and
- A matrix b such that b * a = e, i.e. b is a product of elementary matrices that transforms a into echelon form.
The kernel of the linear transformation a belonging to Hom_(K)(V, W), returned as a subspace of V.
The row nullspace of the linear transformation a belonging to Hom_(K)(V, W), returned as a subspace of W. This is equal to the kernel of the transpose of a.
The cokernel of the linear transformation a belonging to Hom_(K)(V, W).
The transpose of the matrix a.
> Q := RationalField(); > Q2 := VectorSpace(Q, 2); > Q3 := VectorSpace(Q, 3); > Q4 := VectorSpace(Q, 4); > H23 := Hom(Q2, Q3); > H34 := Hom(Q3, Q4); > x := Q2 ! [ -1, 2 ]; > a := H23 ! [ 1/2, 3, 0, 2/3, 4/5, -1 ]; > a; [1/2 3 0] [2/3 4/5 -1] > Domain(a); Full Vector space of degree 2 over Rational Field > Codomain(a); Full Vector space of degree 3 over Rational Field > x*a; ( 5/6 -7/5 -2) > b := H34 ! [ 2, 0, 1, -1/2, 1, 0, 3/2, 4, 4/5, 6/7, 0, -9/7]; > b; [ 2 0 1 -1/2] [ 1 0 3/2 4] [ 4/5 6/7 0 -9/7] > c := a*b; > c; [ 4 0 5 47/4] [ 4/3 -6/7 28/15 436/105] > x*c; ( -4/3 -12/7 -19/15 -1447/420) > Image(c); Vector space of degree 4, dimension 2 over Rational Field Echelonized basis: ( 1 0 5/4 47/16) ( 0 1 -7/30 -11/40) > Kernel(c); Vector space of degree 2, dimension 0 over Rational Field > Rank(c); 2 > EchelonForm(c); [ 1 0 5/4 47/16] [ 0 1 -7/30 -11/40]