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].

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**

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.

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**.

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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*

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Let W_n be the subset of all M_(n x n)(

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.

- Representing Compositions of Linear Transformations as Matrices
- Proving Properties of Compositions of Linear Transformations
- Three Animal Fables
- When are Squares Triangles?
- Linear Dependence and Independence

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