Let α = {[[1,0][0,0]], [[0,1],[0,0]], [[0,0],[1,0]], [[0,0],[0,1]]} and β = {1, x, x^2}, the standard ordered bases for M_(2x2)(**R**) and P_(2)(**R**) respectively. Let γ = {1}.

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”**.*

- Computing Matrix Representations of Ordered Bases
- Representing Compositions of Linear Transformations as Matrices
- Proving Properties of Compositions of Linear Transformations
- Three Animal Fables
- When are Squares Triangles?

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