Viewing posts tagged proof
Theorem: Let V be a vector space. Let R, S, and T ∈ L(V) be linear transformations. Then:
(a) R(S + T) = RS + RT and (S + T)R = SR + TR
(b) R(ST) = (RS)T
(c) RI = IR = R
(d) c(ST) = (cS)T = S(cT) ∀ c ∈ R
This is based upon the final part of this discussion (2) for children given by Rav Ginzburgh on the relationship of square and triangle numbers in the structure of Torah and his paper, "When Two Triangles Make a Square”.
Theorem: Let V be a vector space, and let S_1 ⊆ S_2 ⊆ V. If S_1 is linearly dependent, then S_2 is linearly dependent.
Corollary: Let V be a vector space, and let S_1 ⊆ S_2 ⊆ V. If S_2 is linearly independent, then S_1 is linearly independent.
Theorem: If given S_1 ⊆ S_2 ⊆ V: S_1 is linearly dependent then S_2 is linearly dependent we begin by assuming S_1 is linearly dependent. Then for some arbitrary set of vectors u_1,...,u_n in S_2, then for all a_1,...,a_n in R, with at least some a_i ≠ 0, 1 ≤ i ≤ n:
Let W_2 be the set of all 2 x 2 real symmetric matrices such that for all A in M_(2 x 2)(R), A_ij = A_ji; for all 1 ≤ i,j ≤ 2.
Theorem: A subspace W of a vector space V over R is a subset of V which also has the properties that W is closed under addition and scalar multiplication. That is, For all x, y in W, x and y are in V and for any c in R, cx + y is in W.
Let W_n be the subset of all M_(n x n)(R) such that A_ij = A_ji (ie. the set of all real symmetric square matrices).