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How to count from one to one quintillion

Representing Compositions of Linear Transformations as Matrices

Let S:U->V and T:V->W be linear transformations, and let A = [T][α,β] and B = [S][β,γ] with respective bases of U, V, and W given as α = {u_1,..,u_n}, β = {v_1,..,v_m}, and γ = {w_1,..,w_p}; [1 < n,m,p < ∞] ∈ Z+. The product of these two matrices AB = [TS][α,γ].

Proving Properties of Compositions of Linear Transformations

Theorem: Let V be a vector space. Let R, S, and TL(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

Three Animal Fables

A tortoise and a mallard are walking a cobblestone road in the old town. The mallard turns to the tortoise and asks "So, how am I to come out of my shell?" "I'd prefer to duck that question entirely!" replies the tortoise.

When are Squares Triangles?

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.

Linear Dependence and Independence

Theorem: Let V be a vector space, and let S_1S_2V. If S_1 is linearly dependent, then S_2 is linearly dependent.
Corollary: Let V be a vector space, and let S_1S_2V. 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:

Span(S) is the set of all 2 x 2 Symmetric Matrices

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.

Symmetric Matrices as a Subspace of all Square Matrices

Theorem: 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).

Il Y A

The IL Y A double-sided video, 12-channel sound installation, mixes live video from its two sides so you see through its opaque wall as if it were a glass window. IL Y A transforms what you see of the other side: your gesture transmutes the other, conjures the other’s body. Your movement distends what you see of the other side like smoke or other pseudo-physical material. The effect is symmetrical – any movement by the other reshapes your image as well. Over time, the behavior of the installation changes through a field of behaviors staged by the composer, according also to the activities of its visitors.