Viewing posts tagged Linear Algebra

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**

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)(

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