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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].

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.
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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).