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Computing Matrix Representations of Ordered Bases

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}.

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.